The moment-generating function of a normally distributed

Let be a random variable with \(p(y)\) given in the table below. 1 2 3 4 a. Give the distribution function, \(F(y)\). Be sure to specify the value of \(F(y)\) for all y, \(-\infty<y<\infty\). b. Sketch the distribution function given in part (a). Equation Transcription: Text Transcription: p(y) F(y) F(y) -infinity<y<infinity

Problem 2E A box contains five keys, only one of which will open a lock. Keys are randomly selected and tried, one at a time, until the lock is opened (keys that do not work are discarded before another is tried). Let Y be the number of the trial on which the lock is opened. a Find the probability function for Y . b Give the corresponding distribution function. c What is P(Y < 3)? P(Y ? 3)? P(Y = 3)? d If Y is a continuous random variable, we argued that, for all?? < a < ?, P(Y = a) = 0. Do any of your answers in part (c) contradict this claim? Why?

A Bernoulli random variable is one that assumes only two values, 0 and 1 with \(p(1)=p\) and \(p(0)=1-p \equiv q\) a. Sketch the corresponding distribution function. b. Show that this distribution function has the properties given in Theorem Equation Transcription: Text Transcription: p(1)=p p(0)=1-p identical q

Let Y be a binomial random variable with \(n=1\) and success probability . a. Find the probability and distribution function for . b. Compare the distribution function from part (a) with that in Exercise 4.3(a). What do you conclude? Equation Transcription: Text Transcription: n=1

Suppose that is a random variable that takes on only integer values and has distribution function \(F(y)\). Show that the probability function \(p(y)=P(Y=y)\) is given by \(p(y)=\left\{\begin{array}{ll} F(1), & y=1, \\ F(y)-F(y-1), & y=2,3, \ldots . \end{array}\right. \) Equation Transcription: Text Transcription: F(y) p(y)=P(Y=y) p(y)=F(1), y=1 p(y)=F(1)-F(y-1) y=2

Suppose that Y has a beta distribution with parameters \(\alpha\) and \(\beta\). a If ???? is any positive or negative value such that \(\alpha+a>0\), show that \(E\left(Y^{a}\right)=\frac{\Gamma(\alpha+a) \Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\alpha+\beta+a)}\). b Why did your answer in part (a) require that \(\alpha+a>0\)? c Show that, with \(a=1\), the result in part (a) gives \(E(Y)=\alpha /(\alpha+\beta)\). d Use the result in part (a) to give an expression for \(E(\sqrt{Y})\). What do you need to assume about \(\alpha\)? e Use the result in part (a) to give an expression for \(E(1 / Y)\), \(E(1 / \sqrt{Y})\), and \(E\left(1 / Y^{2}\right)\). What do you need to assume about \(\alpha\) in each case? Equation Transcription: Text Transcription: Alpha beta alpha.+a>0 E(Y^a)=Gamma(alpha+a)Gamma(alpha+beta) over Gamma(alpha)(alpha+beta+a) alpha+a>0 a=1 E(Y)=alpha/(alpha+beta) E(sqrt Y) alpha E(1/Y) E(1/sqrt Y) E(1/Y^2) Alpha

Consider a random variable with a geometric distribution (Section 3.5); that is, \(p(y)=q^{y-1} p\), \(y=1\), 2, 3,...., \(0<p<1\) a. Show that has distribution function \(F(y)\) such that \(F(i)=1-q^{i}\), \(i=0\), 1, 2,... and that, in general, \(F(y)=\left\{\begin{array}{ll} 0, & y<0 \\ 1-q^{i}, & i \leq y<i+1, \quad \text { for } i=0,1,2, \ldots . \end{array}\right.\) b. Show that the preceding cumulative distribution function has the properties given in Theorem . Equation Transcription: Text Transcription: p(y)=q^y-1 p y=1 0<p<1 F(y) F(i)=1-q^1 i=0 F(y)=0 y<0 F(y)=1-q^i i</=y<i+1 i=0

Suppose that has density function \(f(y)=\left\{\begin{array}{ll} k y(1-y), & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the value of that makes a probability density function. b Find \(P(.4 \leq Y \leq 1)\). c Find \(P(.4 \leq Y<1)\). d Find \(P(Y \leq .4 \mid Y \leq .8)\). e Find \(P(Y<.4 \mid Y<.8)\). Equation Transcription: Text Transcription: f(y)={_0, elsewhere. ^ky(1-y), 0y1, P(.4</=Y/=1) P(.4</=Y<1) P(Y.</=4|Y.</=8) P(Y<.4|Y<.8)

A random variable has the following distribution function: \(F(y)=P(Y \leq y)=\left\{\begin{array}{ll} 0, & \text { for } y<2 \\ 1 / 8, & \text { for } 2 \leq y<2.5, \\ 3 / 16, & \text { for } 2.5 \leq y<4, \\ 1 / 2 & \text { for } 4 \leq y<5.5, \\ 5 / 8, & \text { for } 5.5<y<6, \\ 11 / 16, & \text { for } 6 \leq y<7, \\ 1, & \text { for } y \geq 7 \end{array}\right. \) a. Is a continuous or discrete random variable? Why? b. What values of are assigned positive probabilities? c. Find the probability function for . d. What is the median, \(\phi_{.5}\), of ? Equation Transcription: ????.5 Text Transcription: Phi_.5 f(y)=P(Y</=y) 0, for y<2, 1/8, for 2</=y<2.5, 3/16, for 2.5</=y<4, 1/2 for 4</=y<5.5, 5/8, for 5.5<y<6, 11/16, for 6</=y<7, 1, for y>/=7.

Refer to the density function given in Exercise . a. Find the .95-quantile, \(\phi_{.95}\), such that \(P\left(Y \leq \phi_{.95}\right)=.95\). b. Find a value \(y_{0}\) so that \(P\left(Y<y_{0}\right)=.95\). c. Compare the values for \(\phi_{.95}\) and \(y_{0}\) that you obtained in parts (a) and (b). Explain the relationship between these two values. Equation Transcription: ????.95 ????.95 Text Transcription: phi_.95 P(Y</=phi_.95)=.95 y_0 P(Y<y_0)=.95 phi_.95 y_0

The length of time to failure (in hundreds of hours) for a transistor is a random variable with distribution function given by \(F(y)=\left\{\begin{array}{ll} 0, & y<0, \\ 1-e^{-y^{2}}, & y \geq 0. \end{array}\right. \) a Show that \(F(y)\) has the properties of a distribution function. b Find the .30-quantile, \(\phi_{.30}\), of Y. c Find \(f(y)\). d Find the probability that the transistor operates for at least 200 hours. e Find \(P(Y>100 \mid Y \leq 200)\). Equation Transcription: ????.30 Text Transcription: F(y)={_1-e^-y^2, y>/=0. ^0, y<0, F(y) phi_.30 f(y) P(Y>100|Y</=200)

A supplier of kerosene has a 150 -gallon tank that is filled at the beginning of each week. His weekly demand shows a relative frequency behavior that increases steadily up to 100 gallons and then levels off between 100 and 150 gallons. If denotes weekly demand in hundreds of gallons, the relative frequency of demand can be modeled by \(f(y)=\left\{\begin{array}{ll} y, & 0 \leq y \leq 1, \\ 1, & 1<y \leq 1.5, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find \(F(y)\). b Find \(P(0 \leq Y \leq .5)\). c Find \(P(.5 \leq Y \leq 1.2)\). Equation Transcription: Text Transcription: f(y)= y, 0</=y</=1, 1, 1<y</=1.5, 0, elsewhere. F(y) P(0</=Y</=.5) P(.5</=Y</=1.2)

A gas station operates two pumps, each of which can pump up to 10,000 gallons of gas in a month. The total amount of gas pumped at the station in a month is a random variable (measured in 10,000 gallons) with a probability density function given by \(f(y)=\left\{\begin{array}{ll} y, & 0<y<1, \\ 2-y, & 1 \leq y<2, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Graph \(f(y)\). b Find \(F(y)\) and graph it. c Find the probability that the station will pump between 8000 and 12,000 gallons in a particular month. d Given that the station pumped more than 10,000 gallons in a particular month, find the probability that the station pumped more than 15,000 gallons during the month. Equation Transcription: Text Transcription: f(y)= y, 0<y<1, 2-y,</=1y<2, 0, elsewhere. f(y) F(y)

Suppose that possesses the density function \(f(y)=\left\{\begin{array}{ll} c y, & 0 \leq y \leq 2, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the value of that makes \(f(y)\) a probability density function. b Find \(F(y)\). c Graph \(f(y)\) and \(F(y)\). d Use \(F(y)\) to find \(P(1 \leq Y \leq 2)\). e Use \(f(y)\) and geometry to find \(P(1 \leq Y \leq 2)\). Equation Transcription: Text Transcription: f(y)= cy, 0</=y</=2, 0, elsewhere. f(y) F(y) f(y) F(y) F(y) P(1</=Y</=2) f(y) P(1</=Y</=2)

As a measure of intelligence, mice are timed when going through a maze to reach a reward of food. The time (in seconds) required for any mouse is a random variable with a density function given by \(f(y)=\left\{\begin{array}{ll} \frac{b}{y^{2}}, & y \geq b, \\ 0, & \text { elsewhere, } \end{array}\right. \) where is the minimum possible time needed to traverse the maze. a Show that \(f(y)\) has the properties of a density function. b Find \(F(y)\) c Find \(P(Y>b+c)\) for a positive constant . d If and are both positive constants such that \(d>c\), find \(P(Y>b+d \mid Y>b+c)\). Equation Transcription: Text Transcription: f(y)= b over y^2, y>/=b, 0, elsewhere, f(y) F(y) P(Y>b+c) d>c P(Y>b+d|Y>b+c)

Let be a binomial random variable with \(n=10\) and \(p=.2\). a Use Table 1, Appendix 3, to obtain \(P(2<Y<5)\) and \(P(2 \leq Y<5)\). Are the probabilities that falls in the intervals (2,5) and [2,5) equal? Why or why not? b Use Table 1 , Appendix 3 , to obtain \(P(2<Y \leq 5)\) and \(P(2 \leq Y \leq 5)\). Are these two probabilities equal? Why or why not? c Earlier in this section, we argued that if is continuous and \(a<b\), then \(P(a<Y<b)=P(a \leq Y<b)\). Does the result in part (a) contradict this claim? Why? Equation Transcription: Text Transcription: n=10 p=.2 P(2<Y<5) P(2</=Y<5) P(2<Y</=5) P(2</=Y</=5) a<b P(a<Y<b)=P(a</=Y<b)

Let possess a density function \(f(y)=\left\{\begin{array}{ll} c(2-y), & 0 \leq y \leq 2, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find . b Find \(F(y)\). c Graph \(f(y)\) and \(F(y)\). d Use \(F(y)\) in part (b) to find \(P(1 \leq Y \leq 2)\). e Use geometry and the graph for \(f(y)\) to calculate \(P(1 \leq Y \leq 2)\). Equation Transcription: Text Transcription: f(y)= c(2-y), 0</=y</=2, 0, elsewhere. F(y) f(y) F(y) F(y) P(1</=Y</=2) f(y) P(1</=Y</=2)

The length of time required by students to complete a one-hour exam is a random variable with a density function given by \(f(y)=\left\{\begin{array}{ll} c y^{2}+y, & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find . b Find \(F(y)\). c Graph \(f(y)\) and \(F(y)\). d Use \(F(y)\) in part (b) to find \(F(-1)\), \(F(0)\) and \(F(1)\). e Find the probability that a randomly selected student will finish in less than half an hour. f Given that a particular student needs at least 15 minutes to complete the exam, find the probability that she will require at least 30 minutes to finish. Equation Transcription: Text Transcription:

Let have the density function given by \(f(y)=\left\{\begin{array}{ll} .2, & -1<y \leq 0, \\ .2+c y, & 0<y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find . b Find \(F(y)\). c Graph \(f(y)\) and \(F(y)\). d Use \(F(y)\) in part (b) to find \(F(-1)\), \(F(0)\) and \(F(1)\). e Find \(P(0 \leq Y \leq .5)\). f Find \(P(Y>.5 \mid Y>.1)\). Equation Transcription: Text Transcription: f(y)= .2, -1<y0, .2+cy, 0<y1, 0, elsewhere. F(y) f(y) F(y) F(y) F(-1) F(0) F(1) P(0</=Y</=.5) P(Y>.5|Y>.1)

If, as in Exercise 4.16, has density function \(f(y)=\left\{\begin{array}{ll} (1 / 2)(2-y), & 0 \leq y \leq 2, \\ 0, & \text { elsewhere, } \end{array}\right. \) find the mean and variance of . Equation Transcription: Text Transcription: f(y)= (1/2)(2-y), 0</=y</=2, 0, elsewhere,

If, as in Exercise has density function \(f(y)=\left\{\begin{array}{ll} .2, & -1<y \leq 0, \\ .2+(1.2) y, & 0<y \leq 1, \\ 0, & \text { elsewhere, } \end{array}\right. \) find the mean and variance of . Equation Transcription: Text Transcription: f(y)= .2, -1<y</=0, .2+(1.2)y, 0<y</=1, 0, elsewhere,

Let the distribution function of a random variable be \(F(y)=\left\{\begin{array}{ll} 0, & y \leq 0, \\ \frac{y}{8}, & 0<y<2, \\ \frac{y^{2}}{16}, & 2 \leq y<4, \\ 1, & y \geq 4 . \end{array}\right. \) a Find the density function of . b Find \(P(1 \leq Y \leq 3)\). c Find \(P(Y \geq 1.5)\). d Find \(P(Y \geq 1 \mid Y \leq 3)\). Equation Transcription: Text Transcription: F(y)= 0, y</=0, u over 8, 0<y<2, y^2 over 16, 2</=y<4, 1, y>/=4. P(1</=Y</=3) P(Y>/=1.5) P(Y>/=1|Y</=3)

Prove Theorem .

If is a continuous random variable with density function \(f(y)\), use Theorem to prove that \(\sigma^{2}=V(Y)=E\left(Y^{2}\right)-[E(Y)]^{2}\). Equation Transcription: Text Transcription: f(y) sigma^2=V(Y)=E(Y^2)-[E(Y)]^2

If, as in Exercise 4.19, has distribution function \(F(y)=\left\{\begin{array}{ll} 0, & y \leq 0, \\ \frac{y}{8}, & 0<y<2, \\ \frac{y^{2}}{16}, & 2 \leq y<4, \\ 1, & y \geq 4, \end{array}\right. \) find the mean and variance of . Equation Transcription: Text Transcription: F(y)= 0, y</=0, y over 8, 0<y<2, y^2 over 16, 2</=y<4, 1, y>/=4,

If, as in Exercise 4.17, has density function \(f(y)=\left\{\begin{array}{ll} (3 / 2) y^{2}+y, & 0 \leq y \leq 1, \\ 0, & \text { elsewhere, } \end{array}\right. \) find the mean and variance of . Equation Transcription: Text Transcription: f(y)= (3/2)y^2+y, 0</=y</=1, 0, elsewhere,

If is a continuous random variable with mean and variance and and are constants, use Theorem to prove the following: a \(E(a Y+b)=a E(Y)+b=a \mu+b\). b \(V(a Y+b)=a^{2} V(Y)=a^{2} \sigma^{2}\). Equation Transcription: Text Transcription: E(aY+b)=aE(Y)+b=a mu+b V(aY+b)=a^2V(Y)=a^2 sigma^2

For certain ore samples, the proportion of impurities per sample is a random variable with density function given in Exercise 4.21. The dollar value of each sample is \(W=5-.5 Y\). Find the mean and variance of . Equation Transcription: Text Transcription: W=5-.5Y

The proportion of time per day that all checkout counters in a supermarket are busy is a random variable with density function \(f(y)=\left\{\begin{array}{ll} c y^{2}(1-y)^{4}, & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the value of that makes \(f(y)\) a probability density function. b Find \(E(Y)\). Equation Transcription: Text Transcription:

The temperature at which a thermostatically controlled switch turns on has probability density function given by \(f(y)=\left\{\begin{array}{ll} 1 / 2, & 59 \leq y \leq 61, \\ 0, & \text { elsewhere. } \end{array}\right. \) Find \(E(Y)\) and \(V(Y)\). Equation Transcription: Text Transcription: f(y)= 1/2, 59</=y</=61, 0, elsewhere. E(Y) V(Y)

The of water samples from a specific lake is a random variable with probability density function given by \(f(y)=\left\{\begin{array}{ll} (3 / 8)(7-y)^{2}, & 5 \leq y \leq 7, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find \(E(Y)\) and \(V(Y)\). b Find an interval shorter than in which at least three-fourths of the measurements must lie. c Would you expect to see a pH measurement below very often? Why? Equation Transcription: Text Transcription: f(y)= (3/8)(7-y)^2, 5</=y</=7, 0, elsewhere. E(Y) V(Y)

Weekly CPU time used by an accounting firm has probability density function (measured in hours) given by \(f(y)=\left\{\begin{array}{ll} (3 / 64) y^{2}(4-y), & 0 \leq y \leq 4, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the expected value and variance of weekly CPU time. b The CPU time costs the firm per hour. Find the expected value and variance of the weekly cost for CPU time. Equation Transcription: Text Transcription: f(y)= (3/64)y^2(4-y), 0</=y</=4, 0, elsewhere.

The proportion of time that an industrial robot is in operation during a 40 -hour week is a random variable with probability density function \(f(y)=\left\{\begin{array}{ll} 2 y, & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find \(E(Y)\) and \(V(Y)\). b For the robot under study, the profit for a week is given by \(X=200Y-60\). Find \(\) and \(V(X)\). c Find an interval in which the profit should lie for at least of the weeks that the robot is in use. Equation Transcription: Text Transcription: f(y)= 2y, 0</=y</=1, E(Y) V(Y) X=200Y-60 E(X) V(X)

Suppose that is a continuous random variable with density \(f(y)\) that is positive only if \(y \geq 0\). If \(F(y)\) is the distribution function, show that \(E(Y)=\int_{0}^{\infty} y f(y)\) \(d y=\int_{0}^{\infty}[1-F(y)] d y\). [Hint: If \(y>0\), \(y=\int_{0}^{y} d t\), and \(E(Y)=\int_{0}^{\infty} y f(y)\) \(d y=\int_{0}^{\infty}\left\{\int_{0}^{y} d t\right\} f(y) d y\). Exchange the order of integration to obtain the desired result.\(]^{4}\) Equation Transcription: Text Transcription: f(y) y>/=0 F(y) E(Y)=integral_0^infinity yf(y) dy=integral_0^infinity [1=F(y)]dy y>0 y=integral_0^y dt E(Y)=integral_0^infinity yf(y) dy=integral_0^infinity{integral_0^y dt}f(y)dy

Problem 35E If Y is a continuous random variable such that E[(Y ?a)2] < ? for all a, show that E[(Y ?a)2] is minimized when a = E(Y ). [Hint: E[(Y ? a)2] = E({[Y ? E(Y )] + [E(Y ) ? a]}2).]

Daily total solar radiation for a specified location in Florida in October has probability density function given by \(f(y)=\left\{\begin{array}{ll} (3 / 32)(y-2)(6-y), & 2 \leq y \leq 6, \\ 0, & \text { elsewhere, } \end{array}\right. \) with measurements in hundreds of calories. Find the expected daily solar radiation for October. Equation Transcription: Text Transcription: f(y)= (3/32)(y-2)(6-y), 2</=y</=6, 0, elsewhere,

If is a continuous random variable with density function \(f(y)\) that is symmetric about 0 (that is, \(f(y)=f(-y)\) for all ) and exists, show that \(E(Y)=0\). [Hint: \(E(Y)=\int_{-\infty}^{0} y f(y)+\int_{0}^{\infty} y f(y) d y\) Make the change of variable \(w=-y\) in the first integral.] Equation Transcription: Text Transcription: f(y) f(y)=f(-y) E(Y) E(Y)=0 E(Y)=integral_-infinity^0 yf(y)+integral_0^infinity yf(y)dy w=-y

Problem 36E Is the result obtained in Exercise 4.35 also valid for discrete random variables? Why? Reference If Y is a continuous random variable such that E[(Y ?a)2] < ? for all a, show that E[(Y ?a)2] is minimized when a = E(Y ). [Hint: E[(Y ? a)2] = E({[Y ? E(Y )] + [E(Y ) ? a]}2).]

Problem 40E Suppose that three parachutists operate independently as described in Exercise 4.39. What is the probability that exactly one of the three lands past the midpoint between A and B? Reference If a parachutist lands at a random point on a line between markers A and B, find the probability that she is closer to A than to B. Find the probability that her distance to A is more than three times her distance to B.

Problem 41E A random variable Y has a uniform distribution over the interval (?1, ?2). Derive the variance of Y .

Problem 39E If a parachutist lands at a random point on a line between markers A and B, find the probability that she is closer to A than to B. Find the probability that her distance to A is more than three times her distance to B.

Problem 42E The median of the distribution of a continuous random variable Y is the value ?.5 such that P(Y ? ?.5) = 0.5. What is the median of the uniform distribution on the interval (?1, ?2)?

Problem 38E Suppose that Y has a uniform distribution over the interval (0, 1). a Find F(y). b Show that P(a ? Y ? a + b), for a ? 0, b ? 0, and a + b ? 1 depends only upon the value of b.

The change in depth of a river from one day to the next, measured (in feet) at a specific location, is a random variable Y with the following density function: \(f(y)=\left\{\begin{array}{ll} k, & -2 \leq y \leq 2 \\ 0, & \text { elsewhere. } \end{array}\right. \) a Determine the value of ????. b Obtain the distribution function for ???? . Equation Transcription: Text Transcription: f(y)= k, -2</=y</=2 0, elsewhere.

Problem 45E Upon studying low bids for shipping contracts, a microcomputer manufacturing company finds that intrastate contracts have low bids that are uniformly distributed between 20 and 25, in units of thousands of dollars. Find the probability that the low bid on the next intrastate shipping contract a is below $22,000. b is in excess of $24,000.

Problem 47E The failure of a circuit board interrupts work that utilizes a computing system until a new board is delivered. The delivery time, Y , is uniformly distributed on the interval one to five days. The cost of a board failure and interruption includes the fixed cost c0 of a new board and a cost that increases proportionally to Y 2. If C is the cost incurred, C = c0 + c1Y 2. a Find the probability that the delivery time exceeds two days. b In terms of c0 and c1, find the expected cost associated with a single failed circuit board.

Problem 46E Refer to Exercise 4.45. Find the expected value of low bids on contracts of the type described there. Reference Upon studying low bids for shipping contracts, a microcomputer manufacturing company finds that intrastate contracts have low bids that are uniformly distributed between 20 and 25, in units of thousands of dollars. Find the probability that the low bid on the next intrastate shipping contract a is below $22,000. b is in excess of $24,000.

Problem 43E A circle of radius r has area A = ? r 2. If a random circle has a radius that is uniformly distributed on the interval (0, 1), what are the mean and variance of the area of the circle?

Problem 48E If a point is randomly located in an interval (a, b) and if Y denotes the location of the point, then Y is assumed to have a uniform distribution over (a, b). A plant efficiency expert randomly selects a location along a 500-foot assembly line from which to observe the work habits of the workers on the line. What is the probability that the point she selects is a within 25 feet of the end of the line? b within 25 feet of the beginning of the line? c closer to the beginning of the line than to the end of the line?

Problem 49E A telephone call arrived at a switchboard at random within a one-minute interval. The switch board was fully busy for 15 seconds into this one-minute period. What is the probability that the call arrived when the switchboard was not fully busy?

Problem 50E Beginning at 12:00 midnight, a computer center is up for one hour and then down for two hours on a regular cycle. A person who is unaware of this schedule dials the center at a random time between 12:00 midnight and 5:00 A.M. What is the probability that the center is up when the person’s call comes in?

The cycle time for trucks hauling concrete to a highway construction site is uniformly distributed over the interval 50 to 70 minutes. What is the probability that the cycle time exceeds 65 minutes if it is known that the cycle time exceeds 55 minutes?

Problem 52E Refer to Exercise 4.51. Find the mean and variance of the cycle times for the trucks. Reference The cycle time for trucks hauling concrete to a high way construction site is uniformly distributed over the interval 50to 70minutes. What is the probability that the cycle time exceeds 65 minutes if it is known that the cycle time exceeds 55 minutes?

Problem 53E The number of defective circuit boards coming off a soldering machine follows a Poisson distribution. During a specific eight-hour day, one defective circuit board was found. a Find the probability that it was produced during the first hour of operation during that day. b Find the probability that it was produced during the last hour of operation during that day. c Given that no defective circuit boards were produced during the first four hours of operation, find the probability that the defective board was manufactured during the fifth hour.

Problem 55E Refer to Exercise 4.54. Suppose that measurement errors are uniformly distributed between ?0.02 to +0.05 ?s. a What is the probability that a particular arrival-time measurement will be accurate to within 0.01 ?s? b Find the mean and variance of the measurement errors. Reference In using the triangulation method to determine the range of an acoustic source, the test equipment must accurately measure the time at which the spherical wave front arrives at a receiving sensor. According to Perruzzi and Hilliard (1984), measurement errors in these times can be modeled as possessing a uniform distribution from ?0.05 to +0.05 ?s (microseconds). a What is the probability that a particular arrival-time measurement will be accurate to within 0.01 ?s? b Find the mean and variance of the measurement errors.

Refer to Example 4.7. Find the conditional probability that a customer arrives during the last 5 minutes of the 30-minute period if it is known that no one arrives during the first 10 minutes of the period.

Problem 57E According to Zimmels (1983), the sizes of particles used in sedimentation experiments often have a uniform distribution. In sedimentation involving mixtures of particles of various sizes, the larger particles hinder the movements of the smaller ones. Thus, it is important to study both the mean and the variance of particle sizes. Suppose that spherical particles have diameters that are uniformly distributed between .01 and .05 centimeters. Find the mean and variance of the volumes of these particles. (Recall that the volume of a sphere is (4/3)?r 3.)

Problem 54E In using the triangulation method to determine the range of an acoustic source, the test equipment must accurately measure the time at which the spherical wave front arrives at a receiving sensor. According to Perruzzi and Hilliard (1984), measurement errors in these times can be modeled as possessing a uniform distribution from ?0.05 to +0.05 ?s (microseconds). a What is the probability that a particular arrival-time measurement will be accurate to within 0.01 ?s? b Find the mean and variance of the measurement errors.

A normally distributed random variable has density function \(f(y)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(y-\mu)^{2} /\left(2 \sigma^{2}\right)}\), \(-\infty<y<\infty\). Using the fundamental properties associated with any density function, argue that the parameter \(\sigma\) must be such that \(\sigma>0\). Equation Transcription: Text Transcription: f(y)=12e-(y-)2/(22) -infinity<y<infinity sigma sigma>0

Problem 61E What is the median of a normally distributed random variable with mean ? and standard deviation ??

Problem 59E If Z is a standard normal random variable, find the value z0 such that a P ( Z > z 0 ) = .5. b P ( Z < z 0 ) = .8643. c P (?z0 < Z < z0) = .90. d P (?z0 < Z < z0) = .99.

A company that manufactures and bottles apple juice uses a machine that automatically fills 16-ounce bottles. There is some variation, however, in the amounts of liquid dispensed into the bottles that are filled. The amount dispensed has been observed to be approximately normally distributed with mean 16 ounces and standard deviation 1 ounce. a Use Table 4, Appendix 3, to determine the proportion of bottles that will have more than 17 ounces dispensed into them. b Applet Exercise Use the applet Normal Probabilities to obtain the answer to part (a).

The weekly amount of money spent on maintenance and repairs by a company was observed, over a long period of time, to be approximately normally distributed with mean $400 and standard deviation $20. If $450 is budgeted for next week, what is the probability that the actual costs will exceed the budgeted amount? a Answer the question, using Table 4, Appendix 3. b Applet Exercise Use the applet Normal Probabilities to obtain the answer. c Why are the labeled values different on the two horizontal axes?

Problem 62E If Z is a standard normal random variable, what is a P ( Z 2 < 1)? b P ( Z 2 < 3.84146)?

In Exercise 4.64, how much should be budgeted for weekly repairs and maintenance to provide that the probability the budgeted amount will be exceeded in a given week is only .1?

Use Table 4, Appendix 3, to find the following probabilities for a standard normal random variable Z: a \(\mathrm{P}(0 \leq Z \leq 1.2)\) b \(\mathrm{P}(-.9 \leq \mathrm{Z} \leq 0)\) c \(P(.3 \leq Z \leq 1.56)\) d \(P(-.2 \leq Z \leq .2)\) e \(P(?1.56 ? Z ? ?.2)\) f Applet Exercise Use the applet Normal Probabilities to obtain \(P(0 \leq Z \leq 1.2)\). Why are the values given on the two horizontal axes identical? Equation Transcription: Text Transcription: P(0</=Z</=1.2) P(-.9</=Z</=0) P(.3</=Z</=1.56) P(80.0000<Y<90.0000)=P(0.50<Z<1.50)=0.2417 Prob=0.2417 P(-.2</=Z</=.2) P(-1.56</=Z</=-.2) P(0</=Z</=1.2)

In Exercise 4.66, what should the mean diameter be in order that the fraction of bearings scrapped be minimized?

Refer to Exercise 4.68. If students possessing a GPA less than 1.9 are dropped from college, what percentage of the students will be dropped?

Problem 72E One method of arriving at economic forecasts is to use a consensus approach. A forecast is obtained from each of a large number of analysts; the average of these individual forecasts is the consensus forecast. Suppose that the individual 1996 January prime interest–rate forecasts of all economic analysts are approximately normally distributed with mean 7% and standard deviation 2.6%. If a single analyst is randomly selected from among this group, what is the probability that the analyst’s forecast of the prime interest rate will a exceed 11%? b be less than 9%?

Problem 71E Wires manufactured for use in a computer system are specified to have resistances between .12 and .14 ohms. The actual measured resistances of the wires produced by company A have a normal probability distribution with mean .13 ohm and standard deviation .005 ohm. a What is the probability that a randomly selected wire from company A’s production will meet the specifications? b If four of these wires are used in each computer system and all are selected from company A, what is the probability that all four in a randomly selected system will meet the specifications?

A machining operation produces bearings with diameters that are normally distributed with mean 3.0005 inches and standard deviation .0010 inch. Specifications require the bearing diameters to lie in the interval \(3.000 \pm .0020\) inches. Those outside the interval are considered scrap and must be remachined. With the existing machine setting, what fraction of total production will be scrap? a Answer the question, using Table 4, Appendix 3. b Applet Exercise Obtain the answer, using the applet Normal Probabilities. Equation Transcription: Text Transcription: 3.000+/-.0020

Problem 73E The width of bolts of fabric is normally distributed with mean 950 mm (millimeters) and standard deviation 10 mm. a What is the probability that a randomly chosen bolt has a width of between 947 and 958mm? b What is the appropriate value for C such that a randomly chosen bolt has a width less than C with probability .8531?

Problem 74E Scores on an examination are assumed to be normally distributed with mean 78 and variance 36. a What is the probability that a person taking the examination scores higher than 72? b Suppose that students scoring in the top 10% of this distribution are to receive an A grade. What is the minimum score a student must achieve to earn an A grade? c What must be the cutoff point for passing the examination if the examiner wants only the top 28.1% of all scores to be passing? d Approximately what proportion of students have scores 5 or more points above the score that cuts off the lowest 25%? e Applet Exercise Answer parts (a)–(d), using the applet Normal Tail Areas and Quantiles. f If it is known that a student’s score exceeds 72, what is the probability that his or her score exceeds 84?

Refer to Exercise 4.68. Suppose that three students are randomly selected from the student body. What is the probability that all three will possess a GPA in excess of 3.0?

Problem 75E A soft-drink machine can be regulated so that it discharges an average of ? ounces per cup. If the ounces of fill are normally distributed with standard deviation 0.3 ounce, give the setting for ?so that 8-ounce cups will overflow only 1% of the time.

Problem 79E Show that the normal density with parameters ? and ? has inflection points at the values ? – ?and ? + ?. (Recall that an inflection point is a point where the curve changes direction from concave up to concave down, or vice versa, and occurs when the second derivative changes sign. Such a change in sign may occur when the second derivative equals zero.)

Problem 77E The SAT and ACT college entrance exams are taken by thousands of students each year. The mathematics portions of each of these exams produce scores that are approximately normally distributed. In recent years, SAT mathematics exam scores have averaged 480 with standard deviation 100. The average and standard deviation for ACT mathematics scores are 18 and 6, respectively. a An engineering school sets 550 as the minimum SAT math score for new students. What percentage of students will score below 550 in a typical year? b What score should the engineering school set as a comparable standard on the ACT math test?

Problem 80E Assume that Y is normally distributed with mean ? and standard deviation ?. After observing a value of Y, a mathematician constructs a rectangle with length L = |Y | and width W = 3|Y |. Let A denote the area of the resulting rectangle. What is E( A)?

Problem 78E Show that the maximum value of the normal density with parameters ? and ? is 1/(? ? 2?) and occurs when y = ?.

The grade point averages (GPAs) of a large population of college students are approximately normally distributed with mean 2.4 and standard deviation .8. What fraction of the students will possess a GPA in excess of 3.0? a Answer the question, using Table 4, Appendix 3. b Applet Exercise Obtain the answer, using the applet Normal Tail Areas and Quantiles.

Problem 76E The machine described in Exercise 4.75 has standard deviation ? that can be fixed at certain levels by carefully adjusting the machine. What is the largest value of ? that will allow the actual amount dispensed to fall within 1 ounce of the mean with probability at least .95? Reference A soft-drink machine can be regulated so that it discharges an average of ? ounces per cup. If the ounces of fill are normally distributed with standard deviation 0.3 ounce, give the setting for ?so that 8-ounce cups will overflow only 1% of the time.

a If \(\alpha>0\), \(\Gamma(\alpha)\) is defined by \(\Gamma(\alpha)=\int_{0}^{\infty} y^{\alpha-1} e^{-y} d y\), show that . *b If \(\alpha>1\), integrate by parts to prove that \(\Gamma(\alpha)=(\alpha-1) \Gamma(\alpha-1)\). Equation Transcription: Text Transcription: alpha>0 Gamma(alpha) Gamma(alpha)=0y-1e-ydy Gamma(1)=1 alpha>1 Gamma(alpha)=(alpha-1)(alpha-1)

Use the results obtained in Exercise to prove that if is a positive integer, then \(\Gamma n=(n-1)\)!. What are the numerical values of \(\Gamma(2)\), \(\Gamma(4)\) and \(\Gamma(7)\)? Equation Transcription: Text Transcription: Gamman=(n-1) Gamma(2) Gamma(4) Gamma(7)

Applet Exercise Use the applet Comparison of Gamma Density Functions to obtain the results given in Figure .

Problem 86E Applet Exercise When we discussed the ? 2 distribution in this section, we presented (with justification to follow in Chapter 6) the fact that if Y is gamma distributed with ? = n/2 for some integer n, then 2Y/? has a ? 2 distribution. In particular, it was stated that when ? = 1.5 and ? = 4, W = Y /2 has a ? 2 distribution with 3 degrees of freedom. a Use the applet Gamma Probabilities and Quantiles to find P(Y < 3.5). b Use the applet Gamma Probabilities and Quantiles to find P(W < 1.75). [Hint: Recall that the ?2 distribution with ? degrees of freedom is just a gamma distribution with ? = ?/2 and ? = 2.] c Compare your answers to parts (a) and (b).

Problem 87E Applet Exercise Let Y and W have the distributions given in Exercise 4.86. a Use the applet Gamma Probabilities and Quantiles to find the .05-quantile of the distribution of Y . b Use the applet Gamma Probabilities and Quantiles to find the .05-quantile of the ? 2 distribution with 3 degrees of freedom. c What is the relationship between the .05-quantile of the gamma (? = 1.5, ? = 4) distribution and the .05-quantile of the ? 2 distribution with 3 degrees of freedom? Explain this relationship. Reference Applet Exercise When we discussed the ? 2 distribution in this section, we presented (with justification to follow in Chapter 6) the fact that if Y is gamma distributed with ? = n/2 for some integer n, then 2Y/? has a ? 2 distribution. In particular, it was stated that when ? = 1.5 and ? = 4, W = Y /2 has a ? 2 distribution with 3 degrees of freedom. a Use the applet Gamma Probabilities and Quantiles to find P(Y < 3.5). b Use the applet Gamma Probabilities and Quantiles to find P(W < 1.75). [Hint: Recall that the ?2 distribution with ? degrees of freedom is just a gamma distribution with ? = ?/2 and ? = 2.] c Compare your answers to parts (a) and (b).

Problem 85E Applet Exercise Use the applet Comparison of Gamma Density Functions to compare gamma density functions with (? = 1, ? = 1), (? = 1, ? = 2), and (? = 1, ? = 4). a What is another name for the density functions that you observed? b Do these densities have the same general shape? c The parameter ? is a “scale” parameter. What do you observe about the “spread” of these three density functions?

Problem 88E The magnitude of earthquakes recorded in a region of North America can be modeled as having an exponential distribution with mean 2.4, as measured on the Richter scale. Find the probability that an earthquake striking this region will a exceed 3.0 on the Richter scale. b fall between 2.0 and 3.0 on the Richter scale.

Applet Exercise Refer to Exercise . Use the applet Comparison of Gamma Density Functions to compare gamma density functions with \((\alpha=4, \beta=1)\), \((\alpha=40, \beta=1)\) and \(\alpha=80, \beta=1)\). a What do you observe about the shapes of these three density functions? Which are less skewed and more symmetric? b What differences do you observe about the location of the centers of these density functions? c Give an explanation for what you observed in part (b). Equation Transcription: Text Transcription: (alpha=4,beta=1) (alpha=40,beta=1) (alpha=80,beta=1)

Problem 90E Refer to Exercise 4.88. Of the next ten earthquakes to strike this region, what is the probability that at least one will exceed 5.0 on the Richter scale? Reference The magnitude of earthquakes recorded in a region of North America can be modeled as having an exponential distribution with mean 2.4, as measured on the Richter scale. Find the probability that an earthquake striking this region will a exceed 3.0 on the Richter scale. b fall between 2.0 and 3.0 on the Richter scale.

Problem 89E If Y has an exponential distribution and P(Y > 2) = .0821, what is a ? = E(Y )? b P ( Y ? 1.7)?

Problem 93E Historical evidence indicates that times between fatal accidents on scheduled American domestic passenger flights have an approximately exponential distribution. Assume that the mean time between accidents is 44 days. a If one of the accidents occurred on July 1 of a randomly selected year in the study period, what is the probability that another accident occurred that same month? b What is the variance of the times between accidents?

The length of time \(Y\) necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula \(C=100+40 Y+3 Y^{2}\) relates the cost \(C\) of completing this operation to the square of the time to completion. Find the mean and variance of \(C\).

Problem 91E The operator of a pumping station has observed that demand for water during early afternoon hours has an approximately exponential distribution with mean 100 cfs (cubic feet per second). a Find the probability that the demand will exceed 200 cfs during the early afternoon on a randomly selected day. b What water-pumping capacity should the station maintain during early afternoons so that the probability that demand will exceed capacity on a randomly selected day is only .01?

Let be an exponentially distributed random variable with mean . Define a random variable in the following way: \(X=k\) if \(k-1 \leq Y<k\) for \(k=1\), 2,.... a Find \(P(X=k)\)for each \(k=1\), 2,.... b Show that your answer to part (a) can be written as \(P(X=k)=\left(e^{-1 / \beta}\right)^{k-1}\left(1-e^{-1 / \beta}\right)\), \(k=1\), 2,.... and that has a geometric distribution with \(p=\left(1-e^{-1 / \beta}\right)\). Equation Transcription: Text Transcription: X=k k-1</=Y<k k=1 P(X=k) k=1 P(X=k)=(e^-1/beta)^k-1(1-e^-1/beta) k=1 p=(1-e^-1/beta)

Problem 94E One-hour carbon monoxide concentrations in air samples from a large city have an approximately exponential distribution with mean 3.6 ppm (parts per million). a Find the probability that the carbon monoxide concentration exceeds 9 ppm during a randomly selected one-hour period. b A traffic-control strategy reduced the mean to 2.5 ppm. Now find the probability that the concentration exceeds 9 ppm.

Problem 97E A manufacturing plant uses a specific bulk product. The amount of product used in one day can be modeled by an exponential distribution with ? = 4 (measurements in tons). Find the probability that the plant will use more than 4 tons on a given day.

Suppose that a random variable has a probability density function given by \(f(y)=\left\{\begin{array}{ll} k y^{3} e^{-y / 2}, & y>0, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the value of that makes \(f(y)\) a density function. b Does have a \(\chi^{2}\) distribution? If so, how many degrees of freedom? c What are the mean and standard deviation of d Applet Exercise What is the probability that lies within 2 standard deviations of its mean? Equation Transcription: Text Transcription: f(y)= ky^3 e^-y/2, y>0, 0, elsewhere. f(y) chi^2

Problem 98E Consider the plant of Exercise 4.97. How much of the bulk product should be stocked so that the plant’s chance of running out of the product is only .05? Reference A manufacturing plant uses a specific bulk product. The amount of product used in one day can be modeled by an exponential distribution with ? = 4 (measurements in tons). Find the probability that the plant will use more than 4 tons on a given day.

Problem 102E Applet Exercise Refer to Exercise 4.97. Suppose that the amount of product used in one day has a gamma distribution with ? = 1.5 and ? = 3. a Find the probability that the plant will use more than 4 tons on a given day. b How much of the bulk product should be stocked so that the plant’s chance of running out of the product is only .05? Reference A manufacturing plant uses a specific bulk product. The amount of product used in one day can be modeled by an exponential distribution with ? = 4 (measurements in tons). Find the probability that the plant will use more than 4 tons on a given day.

If \(\lambda>0\) and \(\alpha\) is a positive integer, the relationship between incomplete gamma integrals and sums of Poisson probabilities is given by \(\frac{1}{\Gamma(\alpha)} \int_{\lambda}^{\infty} y^{\alpha-1} e^{-y}\) \(d y=\sum_{x=0}^{\alpha-1} \frac{\lambda^{x} e^{-\lambda}}{x !}\). a If has a gamma distribution with \(\alpha=2\) and \(\beta=1\), find \(P(Y>1)\) by using the preceding equality and Table 3 of Appendix 3 . b Applet Exercise If has a gamma distribution with \(\alpha=2\) and \(\beta=1\), find \(P(Y>1)\) by using the applet Gamma Probabilities. Equation Transcription: Text Transcription: lambda>0 alpha 1 over Gamma(alpha) integral y to infinity y^alpha-1 e^-y dy=sum x=0 alpha-1 lambda^x e^-lambda over x! alpha=2 beta=1 P(Y>1) alpha=2 beta=1 P(Y>1)

Problem 101E Applet Exercise Refer to Exercise 4.88. Suppose that the magnitude of earthquakes striking the region has a gamma distribution with ? = .8 and ? = 2.4. a What is the mean magnitude of earthquakes striking the region? b What is the probability that the magnitude an earthquake striking the region will exceed 3.0 on the Richter scale? c Compare your answers to Exercise 4.88(a). Which probability is larger? Explain. d What is the probability that an earthquake striking the regions will fall between 2.0 and 3.0 on the Richter scale? Reference The magnitude of earthquakes recorded in a region of North America can be modeled as having an exponential distribution with mean 2.4, as measured on the Richter scale. Find the probability that an earthquake striking this region will a exceed 3.0 on the Richter scale. b fall between 2.0 and 3.0 on the Richter scale.

Problem 103E Explosive devices used in mining operations produce nearly circular craters when detonated. The radii of these craters are exponentially distributed with mean 10 feet. Find the mean and variance of the areas produced by these explosive devices.

The lifetime (in hours) of an electronic component is a random variable with density function given by \(f(y)=\left\{\begin{array}{ll} \frac{1}{100} e^{-y / 100}, & y>0, \\ 0, & \text { elsewhere. } \end{array}\right. \) Three of these components operate independently in a piece of equipment. The equipment fails if at least two of the components fail. Find the probability that the equipment will operate for at least 200 hours without failure. Equation Transcription: Text Transcription: f(y)= 1100e-y/100, y>0, 0, elsewhere.

Problem 106E The response times on an online computer terminal have approximately a gamma distribution with mean four seconds and variance eight seconds2. a Write the probability density function for the response times. b Applet Exercise What is the probability that the response time on the terminal is less than five seconds?

Problem 105E Four-week summer rainfall totals in a section of the Midwest United States have approximately a gamma distribution with ? = 1.6 and ? = 2.0. a Find the mean and variance of the four-week rainfall totals. b Applet Exercise What is the probability that the four-week rainfall total exceeds 4 inches?

Problem 107E Refer to Exercise 4.106. a Use Tchebysheff’s theorem to give an interval that contains at least 75% of the response times. b Applet Exercise What is the actual probability of observing a response time in the interval you obtained in part (a)? Reference The response times on an online computer terminal have approximately a gamma distribution with mean four seconds and variance eight seconds2. a Write the probability density function for the response times. b Applet Exercise What is the probability that the response time on the terminal is less than five seconds?

The weekly amount of downtime \(Y\) (in hours) for an industrial machine has approximately a gamma distribution with \(\alpha=3\) and \(\beta=2\). The loss \(L\) (in dollars) to the industrial operation as a result of this downtime is given by \(L=30 Y+2 Y^{2}\). Find the expected value and variance of \(L\).

Problem 108E Annual incomes for heads of household in a section of a city have approximately a gamma distribution with ? = 20 and ? = 1000. a Find the mean and variance of these incomes. b Would you expect to find many incomes in excess of $30,000 in this section of the city? c Applet Exercise What proportion of heads of households in this section of the city have incomes in excess of $30,000?

If has a probability density function given by \(f(y)=\left\{\begin{array}{ll} 4 y^{2} e^{-2 y}, & y>0, \\ 0, & \text { elsewhere, } \end{array}\right. \) obtain \(E(Y)\) and \(V(Y)\) by inspection. Equation Transcription: Text Transcription: f(y)= 4^2e^-2y, y>0, 0, elsewhere, E(Y) V(Y)

Suppose that has a gamma distribution with parameters \(\alpha\) and \(\beta\). a If \(\alpha\) is any positive or negative value such that \(\alpha+a>0\), show that \(E\left(Y^{a}\right)=\frac{\beta^{a} \Gamma(\alpha+a)}{\Gamma(\alpha)}\) b Why did your answer in part (a) require that \(\alpha+a>0\)? c Show that, with \(a=1\), the result in part (a) gives \(E(Y)=\alpha \beta\). d Use the result in part (a) to give an expression for \(E(\sqrt{Y})\). What do you need to assume about \(\alpha\)? e Use the result in part (a) to give an expression for \(E(1 / Y)\), \(E(1 / \sqrt{Y})\) and \(E\left(1 / Y^{2}\right)\). What do you need to assume about in each case \(\alpha\)? Equation Transcription: Text Transcription: alpha beta alpha alpha+a>0 E(Y^a)=beta^a Gamma(alpha+a) over Gamma(alpha) alpha+a>0 alpha=1 E(Y)=alpha beta E(sqrt Y) alpha E(1/Y) E(1/sqrt Y) E(1/Y^2)

Applet Exercise Refer to Exercise 4.113. Use the applet Comparison of Beta Density Functions to compare beta density functions with \((\alpha=1, \beta=1)\), \((\alpha=1, \beta=2)\) and \((\alpha=2, \beta=1)\). a What have we previously called the beta distribution with \((\alpha=1, \beta=1)\)? b Which of these beta densities is symmetric? c Which of these beta densities is skewed right? d Which of these beta densities is skewed left? *e In Chapter 6 we will see that if is beta distributed with parameters \(\alpha\) and \(\beta\), then \(Y^{*}=1-Y\) has a beta distribution with parameters \(\alpha^{*}=\beta\) and \(\beta^{*}=\alpha\). Does this explain the differences in the graphs of the beta densities? Equation Transcription: Text Transcription: (alpha=1,beta=1) (alpha=1,beta=2) (alpha=2,beta=1) (alpha=1,beta=1) alpha beta Y*=1-Y alpha*=beta beta*=alpha

Applet Exercise Use the applet Comparison of Beta Density Functions to obtain the results given in Figure

Problem 116E Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (? = 1.5, ? = 7), (? = 2.5, ? = 7), and (? = 3.5, ? = 7). a Are these densities symmetric? Skewed left? Skewed right? b What do you observe as the value of ? gets closer to 7? c Graph some more beta densities with ? > 1, ? > 1, and ? < ?. What do you conjecture about the shape of beta densities when both ? > 1, ? >1, and ? < ??

Problem 115E Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (? = 2, ? = 2), (? = 3, ? = 3), and (? = 9, ? = 9). a What are the means associated with random variables with each of these beta distributions? b What is similar about these densities? c How do these densities differ? In particular, what do you observe about the “spread” of these three density functions? d Calculate the standard deviations associated with random variables with each of these beta densities. Do the values of these standard deviations explain what you observed in part (c)? Explain. e Graph some more beta densities with ? = ?. What do you conjecture about the shape of beta densities with ? = ??

Suppose that has a \(\chi^{2}\) distribution with degrees of freedom. Use the results in Exercise in your answers to the following. These results will be useful when we study the and distributions in Chapter 7 . a Give an expression for \(E\left(Y^{a}\right)\) if \(v>-2 a\). b Why did your answer in part (a) require that \(v>-2 a\)? c Use the result in part (a) to give an expression for \(E(\sqrt{Y})\). What do you need to assume about ? d Use the result in part (a) to give an expression for \(E(1 / Y)\), \(E(1 / \sqrt{Y})\), and \(E\left(1 / Y^{2}\right)\). What do you need to assume about in each case? Equation Transcription: Text Transcription: chi^2 E(Y^a) v>-2a v>-2a E(sqrt Y) E(1/Y) E(1/sqrt Y) E(1/Y^2)

Problem 118E Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (? = .3, ? = 4), (? = .3, ? = 7), and (? = .3, ? = 12). a Are these densities symmetric? Skewed left? Skewed right? b What do you observe as the value of ? gets closer to 12? c Which of these beta distributions gives the highest probability of observing a value larger than 0.2? d Graph some more beta densities with? < 1 and? > 1. What do you conjecture about the shape of beta densities with ? < 1 and? > 1?

Problem 117E Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (? = 9, ? = 7), (? = 10, ? = 7), and (? = 12, ? = 7). a Are these densities symmetric? Skewed left? Skewed right? b What do you observe as the value of ? gets closer to 12? c Graph some more beta densities with ? > 1, ? > 1, and ? > ?. What do you conjecture about the shape of beta densities with ? > ? and both ? > 1 and? > 1?

Problem 119E Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (? = 4, ? = 0.3), (? = 7, ? = 0.3), and (? = 12, ? = 0.3). a Are these densities symmetric? Skewed left? Skewed right? b What do you observe as the value of ? gets closer to 12? c Which of these beta distributions gives the highest probability of observing a value less than 0.8? d Graph some more beta densities with? > 1 and? < 1. What do you conjecture about the shape of beta densities with ? > 1 and? < 1?

Problem 121E Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (? = 0.5, ? = 0.7), (? = 0.7, ? = 0.7), and (? = 0.9, ? = 0.7). a What is the general shape of these densities? b What do you observe as the value of ? gets larger?

Problem 122E Applet Exercise Beta densities with ? < 1 and ? < 1 are difficult to display because of scaling/resolution problems. a Use the applet Beta Probabilities and Quantiles to compute P(Y > 0.1) if Y has a beta distribution with (? = 0.1, ? = 2). b Use the applet Beta Probabilities and Quantiles to compute P(Y < 0.1) if Y has a beta distribution with (? = 0.1, ? = 2). c Based on your answer to part (b), which values of Y are assigned high probabilities if Y has a beta distribution with (? = 0.1, ? = 2)? d Use the applet Beta Probabilities and Quantiles to compute P(Y < 0.1) if Y has a beta distribution with (? = 0.1, ? = 0.2). e Use the applet Beta Probabilities and Quantiles to compute P(Y > 0.9) if Y has a beta distribution with (? = 0.1, ? = 0.2). f Use the applet Beta Probabilities and Quantiles to compute P(0.1 < Y < 0.9) if Y has a beta distribution with (? = .1, ? = 0.2). g Based on your answers to parts (d), (e), and (f ), which values of Y are assigned high probabilities if Y has a beta distribution with (? = 0.1, ? = 0.2)?

The percentage of impurities per batch in a chemical product is a random variable with density function \(f(y)=\left\{\begin{array}{ll} 12 y^{2}(1-y) . & 0 \leq y \leq 1, \\ 0 . & \text { elsewhere. } \end{array}\right. \) A batch with more than impurities cannot be sold. a Integrate the density directly to determine the probability that a randomly selected batch cannot be sold because of excessive impurities. b Applet Exercise Use the applet Beta Probabilities and Quantiles to find the answer to part (a). Equation Transcription: Text Transcription: f(y)= 12y^2(1-y), 0</=y</=1, 0, elsewhere.

The weekly repair cost for a machine has a probability density function given by \(f(y)=\left\{\begin{array}{ll} 3(1-y)^{2}, & 0<y<1, \\ 0, & \text { elsewhere, } \end{array}\right. \) with measurements in hundreds of dollars. How much money should be budgeted cach week to for repair costs so that the actual cost will exceed the budgeted amount only of the time? Equation Transcription: Text Transcription: f(y)= 3(1-y)^2, 0</=y</=1, 0, elsewhere,

Problem 127E Verify that if Y has a beta distribution with ? = ? = 1, then Y has a uniform distribution over (0, 1). That is, the uniform distribution over the interval (0, 1) is a special case of a beta distribution.

Refer to Exercise . Find the mean and variance of the percentage of impurities in a randomly selected batch of the chemical.

Problem 120E In Chapter 6 we will see that if Y is beta distributed with parameters ? and ?, then Y ? = 1 – Y has a beta distribution with parameters ? ? = ? and ? ? = ?. Does this explain the differences and similarities in the graphs of the beta densities in Exercises 4.118 and 4.119? Reference Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (? = 4, ? = 0.3), (? = 7, ? = 0.3), and (? = 12, ? = 0.3). a Are these densities symmetric? Skewed left? Skewed right? b What do you observe as the value of ? gets closer to 12? c Which of these beta distributions gives the highest probability of observing a value less than 0.8? d Graph some more beta densities with? > 1 and? < 1. What do you conjecture about the shape of beta densities with ? > 1 and? < 1? Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (? = .3, ? = 4), (? = .3, ? = 7), and (? = .3, ? = 12). a Are these densities symmetric? Skewed left? Skewed right? b What do you observe as the value of ? gets closer to 12? c Which of these beta distributions gives the highest probability of observing a value larger than 0.2? d Graph some more beta densities with? < 1 and? > 1. What do you conjecture about the shape of beta densities with ? < 1 and? > 1?

The relative humidity , when measured at a location, has a probability density function given by \(f(y)=\left\{\begin{array}{ll} k y^{3}(1-y)^{2}, & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the value of that makes \(f(y)\) a density function. b Applet Exercise Use the applet Beta Probabilities and Quantiles to find a humidity value that is exceeded only of the time. Equation Transcription: Text Transcription: f(y)= ky^3(1-y)^2, 0</=y</=1, 0, elsewhere.

During an eight-hour shift, the proportion of time that a sheet-metal stamping machine is down for maintenance or repairs has a beta distribution with \(\alpha=1\) and \(\beta=2\). That is, \(f(y)=\left\{\begin{array}{ll} 2(1-y), & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) The cost (in hundreds of dollars) of this downtime, due to lost production and cost of maintenance and repair, is given by \(C=10+20 Y+4 Y^{2}\). Find the mean and variance of . Equation Transcription: Text Transcription: alpha=1 beta=2 f(y)= 2(1-y), 0</=y</=1, 0, elsewhere. C=10+20Y+4Y^2

Problem 132E Proper blending of fine and coarse powders prior to copper sintering is essential for uniformity in the finished product. One way to check the homogeneity of the blend is to select many small samples of the blended powders and measure the proportion of the total weight contributed by the fine powders in each. These measurements should be relatively stable if a homogeneous blend has been obtained. a Suppose that the proportion of total weight contributed by the fine powders has a beta distribution with ? = ? = 3. Find the mean and variance of the proportion of weight contributed by the fine powders. b Repeat part (a) if ? = ? = 2. c Repeat part (a) if ? = ? = 1. d Which of the cases—parts (a), (b), or (c)—yields the most homogeneous blending?

Prove that the variance of a beta-distributed random variable with parameters \(\alpha\) and \(\beta\) is \(\sigma^{2}=\frac{\alpha \beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}\). Equation Transcription: Text Transcription: alpha beta sigma^2=alpha beta over (alpha+beta)^2(alpha+beta+1)

Problem 131E Errors in measuring the time of arrival of a wave front from an acoustic source sometimes have an approximate beta distribution. Suppose that these errors, measured in microseconds, have approximately a beta distribution with ? = 1 and ? = 2. a What is the probability that the measurement error in a randomly selected instance is less than .5 ?s? b Give the mean and standard deviation of the measurement errors.

The proportion of time per day that all checkout counters in a supermarket are busy is a random variable with a density function given by \(f(y)=\left\{\begin{array}{ll} c y^{2}(1-y)^{4}, & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the value of that makes \(f(y)\) a probability density function. b Find \(E(Y)\). (Use what you have learned about the beta-type distribution. Compare your answers to those obtained in Exercise .) c Calculate the standard deviation of . d Applet Exercise Use the applet Beta Probabilities and Quantiles to find \(P(Y>\mu+2 \sigma)\). Equation Transcription: Text Transcription: f(y)= cy^2(1-y)^4, 0</=y</=1, 0, elsewhere. f(y) E(Y) P(Y>mu+2sigma)

In the text of this section, we noted the relationship between the distribution function of a beta-distributed random variable and sums of binomial probabilities. Specifically, if has a beta distribution with positive integer values for \(\alpha\) and \(\beta\) and \(0<y<1\), \(F(y)=\int_{0}^{y} \frac{t^{\alpha-1}(1-t)^{\beta-1}}{B(\alpha, \beta)}\) \(d t=\sum_{i=\alpha}^{n}\left(\begin{array}{l} n \\ i \end{array}\right) y^{i}(1-y)^{n-1} \), where \(n=\alpha+\beta-1\). a If has a beta distribution with \(\alpha=4\) and \(\beta=7\), use the appropriate binomial tables to find \(P(Y \leq .7)=F(.7)\). b If has a beta distribution with \(\alpha=12\) and \(\beta=14\), use the appropriate binomial tables to find \(P(Y \leq .6)=F(.6)\). c Applet Exercise Use the applet Beta Probabilities and Quantiles to find the probabilities in parts (a) and (b). Equation Transcription: Text Transcription: alpha beta 0<y<1 F(y)=integral 0 to y t^alpha-1(1-t)^beta-1 over B(alpha,beta) dt=sum i=alpha n(_i^n)y^i(1-y)^n-1 n=alpha+beta-1 alpha=4 beta=7 P(Y</=.7)=F(.7) alpha=12 beta=14 P(</=Y.6)=F(.6)

Suppose that the waiting time for the first customer to enter a retail shop after . is a random variable with an exponential density function given by \(f(y)=\left\{\begin{array}{ll} \left(\frac{1}{\theta}\right) e^{-y / \theta}, & y>0, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the moment-generating function for . b Use the answer from part (a) to find \(E(Y)\) and \(V(Y)\). Equation Transcription: Text Transcription: f(y)= (1 over theta)e^-y/theta, y>0 0, elsewhere. E(Y) V(Y)

Suppose that \(Y_{1}\) and \(Y_{2}\) are binomial random variables with parameters \(\left(n, p_{1}\right)\) and \(\left(n, p_{2}\right)\), respectively, where \(p_{1}<p_{2}\). (Note that the parameter is the same for the two variables.) a Use the binomial formula to deduce that \(P\left(Y_{1}=0\right)>P\left(Y_{2}=0\right)\). b Use the relationship between the beta distribution function and sums of binomial probabilities given in Exercise 4.134 to deduce that, if is an integer between 1 and \(n-1\), \(P\left(Y_{1} \leq k\right)=\sum_{i=0}^{k}\left(\begin{array}{l} n \\ i \end{array}\right)\left(p_{1}\right)^{i}\left(1-p_{1}\right)^{n-i}=\int_{p_{1}}^{1} \frac{t^{k}(1-t)^{n-k-1}}{B(k+1, n-k)} d t \). c If is an integer between 1 and \(n-1\), the same argument used in part (b) yields that \(P\left(Y_2\le k\right)=\sum_{i=0}^k\left(\begin{array}{l}n\\ i\end{array}\right)\left(p_2\right)^i\left(1-p_2\right)^{n-i}=\int_{p_2}^1\frac{t^k(1-t)^{n-k-1}}{B(k+1,\ n-k)}dt. \) Show that, if is any integer between 1 and \(n-1\), \(P\left(Y_{1} \leq k\right)>P\left(Y_{2} \leq k\right)\). Interpret this result. Equation Transcription: Text Transcription: Y_1 Y_2 (n,p_1) (n,p_2) p_1<p_2 P(Y_1=0)>P(Y_2=0) n-1 P(Y_1</=k)=sum i=0 k(_i^n)(p_1)^i(1-p_1)^n-i=integral p1 to 1 t^k(1-t)^n-k-1 over B(k+1, n-k)dt n-1 P(Y_</=2k)=sum i=0 k(_i^n)(p_2)^i(1-p_2)^n-i=integral p2 to 1 t^k(1-t)^n-k-1 over B(k+1, n-k)dt n-1 P(Y_1</=k)>P(Y_2</=k)

Example 4.16 derives the moment-generating function for \(Y-\mu\), where ???? is normally distributed with mean \(\mu\) and variance \(\sigma^{2}\). a Use the results in Example 4.16 and Exercise 4.137 to find the moment-generating function for ????. b Differentiate the moment-generating function found in part (a) to show that \(E(Y)=\mu\) and \(V(Y)=\sigma^{2}\). Equation Transcription: Text Transcription: Y-mu mu 2 E(Y)=mu V(Y)=sigma^2

Identify the distributions of the random variables with the following moment-generating functions: a \(m(t)=(1-4 t)^{-2}\). b \(m(t)=1 /(1-3.2 t)\). c \(m(t)=e^{-5 t+6 t^{2}}\). Equation Transcription: Text Transcription: m(t)=(1-4t)^-2 m(t)=1/(1-3.2t) m(t)=e^-5t+6t2

The moment-generating function of a normally distributed random variable, ????, with mean ???? and variance \(\sigma^{2}\) was shown in Exercise 4.138 to be \(m(t)=e^{\mu t+(1 / 2) t^{2} \sigma^{2}}\). Use the result in Exercise 4.137 to derive the moment-generating function of \(???? =\). What is the distribution of ????? Why? Equation Transcription: Text Transcription: 2 m(t)=e^mu t+(1/2)t^2sigma^2 X=-3Y+4

Problem 137E Show that the result given in Exercise 3.158 also holds for continuous random variables. That is, show that, if Y is a random variable with moment-generating function m(t) and U is given by U = aY + b, the moment-generating function of U is etbm(at). If Y has mean ? and variance ? 2, use the moment-generating function of U to derive the mean and variance of U . Reference If Y is a random variable with moment-generating function m(t) and if W is given by W = aY + b, show that the moment-generating function of W is etbm(at).

Problem 142E Refer to Exercises 4.141 and 4.137. Suppose that Y is uniformly distributed on the interval (0, 1) and that a > 0 is a constant. a Give the moment-generating function for Y . b Derive the moment-generating function of W = aY. What is the distribution of W? Why? c Derive the moment-generating function of X = ?aY. What is the distribution of X? Why? d If b is a fixed constant, derive the moment-generating function of V = aY + b. What is the distribution of V? Why? Reference 4.141 If ?1 < ?2, derive the moment-generating function of a random variable that has a uniform distribution on the interval (?1, ?2). 4.137 Show that the result given in Exercise 3.158 also holds for continuous random variables. That is, show that, if Y is a random variable with moment-generating function m(t) and U is given by U = aY + b, the moment-generating function of U is etbm(at). If Y has mean ? and variance ?2, use the moment-generating function of U to derive the mean and variance of U .

Let ???? be a gamma-distributed random variable where \(\alpha\) is a positive integer and \(\beta=1\). The result given in Exercise 4.99 implies that that if \(????>0\), \(\sum_{x=0}^{\alpha-1} \frac{y^{x} e^{-y}}{x !}=P(Y>y)\). Suppose that \(X_{1}\) is Poisson distributed with mean \(\lambda_{1}\) and \(X_{2}\) is Poisson distributed with mean \(\lambda_{2}\), where \(\lambda_{2}>\lambda_{1}\). a Show that \(P\left(X_{1}=0\right)>P\left(X_{2}=0\right)\). b Let ???? be any fixed positive integer. Show that \(P\left(X_{1} \leq k\right)=P\left(Y>\lambda_{1}\right)\) and \(P\left(X_{2} \leq k\right)=P\left(Y>\lambda_{2}\right)\), where ???? is has a gamma distribution with \(\alpha=k+1\) and \(\beta=1\). c Let ???? be any fixed positive integer. Use the result derived in part (b) and the fact that \(\lambda_{2}>\lambda_{1}\) to show that \(P\left(X_{1} \leq k\right)>P\left(X_{2} \leq k\right)\). d Because the result in part (c) is valid for any \(k=1\), 2, 3, . . . and part (a) is also valid, we have established that \(P\left(X_{1} \leq k\right)>P\left(X_{2} \leq k\right)\) for all \(k=0\), 1, 2, . . . . Interpret this result. Equation Transcription: Text Transcription: beta=1 y>0 sum x=0 to alpha-1 y^x e^-y over x!=P(Y>y) X_1 lambda_1 X_2 lambda_2 lambda_2>lambda_1 P(X_1=0)>P(X_2=0) P(X_1</=k)=P(Y>lambda_1) P(X_2</k)=P(Y>lambda_2) alpha=k+1 beta=1 lambda_2>lambda_1 P(X_1</=k)>P(X_2</=k) k=1 P(X_1</=k)>P(X_2</=k)

Problem 141E If ?1 < ?2, derive the moment-generating function of a random variable that has a uniform distribution on the interval (?1, ?2).

Consider a random variable ???? with density function given by \(f(y)=k e^{-y^{2} / 2}\), \(-\infty<y<\infty\). a Find ????. b Find the moment-generating function of ????. c Find \(????(???? )\) and \(????(????)\). Equation Transcription: Text Transcription: f(y)=ke^-y2/2 -infinity <y< infinity E(V) V(Y)

The moment-generating function for the gamma random variable is derived in Example 4.13. Differentiate this moment-generating function to find the mean and variance of the gamma distribution. Find the moment-generating function for a gamma-distributed random variable. \(m(t)=E\left(e^{ty}\right)=\int_0^{\infty}e^{ty}\left[\frac{y^{\alpha-1}e^{-y/\beta}}{\beta^{\alpha}\Gamma(\alpha)}\right]\ dy\) \(=\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\int_0^{\infty}y^{\alpha-1}\exp\left[-y\left(\frac{1}{\beta}-t\right)\right]\ dy\) \(=\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\int_0^{\infty}y^{\alpha-1}\exp\left[\frac{-y}{\beta/(1-\beta t)}\right]\ dy\). Equation Transcription: Text Transcription: m(t)=E(e^ty)=integral 0 to infinity e^ty[y^alpha-1 e^-y/beta over beta^alpha Gamma(alpha)]dy =1 over beta^alpha Gamma(alpha) integral 0 to infinity y^alpha-1 exp[-y(1 over beta -t)]dy =1 over beta^alpha Gamma(alpha) integral 0 to infinity y^alpha-1 exp[-y over /(1-beta t)]dy

Problem 146E A manufacturer of tires wants to advertise a mileage interval that excludes no more than 10% of the mileage on tires he sells. All he knows is that, for a large number of tires tested, the mean mileage was 25,000 miles, and the standard deviation was 4000 miles. What interval would you suggest?

Problem 147E A machine used to fill cereal boxes dispenses, on the average, ? ounces per box. The manufacturer wants the actual ounces dispensed Y to be within 1 ounce of ? at least 75% of the time. What is the largest value of ?, the standard deviation of Y, that can be tolerated if the manufacturer’s objectives are to be met?

A random variable ???? has the density function \(f(y)=\left\{\begin{array}{ll} e^{y}, & y<0, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find \(E\left(e^{3 Y / 2}\right)\). b Find the moment-generating function for ????. c Find \(V(Y)\). Equation Transcription: Text Transcription:

Find \(P(|Y-\mu| \leq 2 \sigma)\) for Exercise 4.16. Compare with the corresponding probabilistic statements given by Tchebysheff’s theorem and the empirical rule. Let ???? possess a density function \(f(y)=\left\{\begin{array}{ll} c(2-y), & 0 \leq y \leq 2, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find ????. b Find \(F(y)\). c Graph \(f(y)\) and \(F(y)\). d Use \(F(y)\) in part (b) to find \(P(1 \leq Y \leq 2)\). e Use geometry and the graph for \(f(y)\) to calculate \(P(1 \leq Y \leq 2)\). Equation Transcription: Text Transcription: P(|Y-mu|</=2sigma) f(y)= c(2-y), 0</=y</=2, 0, elsewhere. F(y) f(y) F(y) F(y) P(1</=Y</=2) f(y) P(1</=Y</=2)

Problem 150E Find P(|Y ? ?| ? 2?) for the exponential random variable. Compare with the corresponding probabilistic statements given by Tchebysheff’s theorem and the empirical rule.

Problem 149E Find P(|Y ? ?| ? 2?) for the uniform random variable. Compare with the corresponding probabilistic statements given by Tchebysheff’s theorem and the empirical rule.

Problem 151E Refer to Exercise 4.92. Would you expect C to exceed 2000 very often? Reference The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula C = 100 + 40Y + 3Y 2 relates the cost C of completing this operation to the square of the time to completion. Find the mean and variance of C.

Refer to Exercise 4.129. Find an interval for which the probability that ???? will lie within it is at least .75. During an eight-hour shift, the proportion of time that a sheet-metal stamping machine is down for maintenance or repairs has a beta distribution with \(\alpha=1\) and \(\beta=2\). That is, \(f(y)=\left\{\begin{array}{ll} 2(1-y), & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) The cost (in hundreds of dollars) of this downtime, due to lost production and cost of maintenance and repair, is given by \(C=10+20 Y+4 Y^{2}\). Find the mean and variance of . Equation Transcription: Text Transcription: alpha=1 beta=2 f(y)= 2(1-y), 0</=y</=1, 0, elsewhere. C=10+20Y+4Y^2

Problem 155E A builder of houses needs to order some supplies that have a waiting time Y for delivery, with a continuous uniform distribution over the interval from 1 to 4 days. Because she can get by without them for 2 days, the cost of the delay is fixed at $100 for any waiting time up to 2 days. After 2 days, however, the cost of the delay is $100 plus $20 per day (prorated) for each additional day. That is, if the waiting time is 3.5 days, the cost of the delay is $100 + $20(1.5) = $130. Find the expected value of the builder’s cost due to waiting for supplies.

Problem 152E Refer to Exercise 4.109. Find an interval that will contain L for at least 89% of the weeks that the machine is in use. Reference The weekly amount of downtime Y (in hours) for an industrial machine has approximately a gamma distribution with ? = 3 and ? = 2. The loss L (in dollars) to the industrial operation as a result of this downtime is given by L = 30Y + 2Y 2. Find the expected value and variance of L.

Problem 154E Suppose that Y is a ? 2 distributed random variable with ? = 7 degrees of freedom. a What are the mean and variance of Y ? b Is it likely that Y will take on a value of 23 or more? c Applet Exercise Use the applet Gamma Probabilities and Quantiles to find P(Y > 23).

Problem 157E The life length Y of a component used in a complex electronic system is known to have an exponential density with a mean of 100 hours. The component is replaced at failure or at age 200 hours, whichever comes first. a Find the distribution function for X, the length of time the component is in use. b Find E( X ).

The duration ???? of long-distance telephone calls (in minutes) monitored by a station is a random variable with the properties that \(P(Y=3)=.2\quad\ \text{ and }\quad\ P(Y=6)=.1.\) Otherwise, ???? has a continuous density function given by \(f(y)=\left\{\begin{array}{ll} (1 / 4) y e^{-y / 2}, & y>0, \\ 0, & \text { elsewhere. } \end{array}\right. \) The discrete points at 3 and 6 are due to the fact that the length of the call is announced to the caller in three-minute intervals and the caller must pay for three minutes even if he talks less than three minutes. Find the expected duration of a randomly selected long-distance call. Equation Transcription: Text Transcription: P(Y=3)=.2 and P(Y=6)=.1. f(y)= (1/4)ye-y/2, y>0, 0, elsewhere.

A random variable ???? has distribution function \(F(y)=\left\{\begin{array}{ll} 0, & \text { if } y<0, \\ y^{2}+0.1, & \text { if } 0 \leq y<0.5, \\ y, & \text { if } 0.5 \leq y<1, \\ 1, & \text { if } y \geq 1. \end{array}\right. \) a Give \(F_{1}(y)\) and \(F_{2}(y)\), the discrete and continuous components of \(F(y)\). b Write \(F(y)\) as \(c_{1} F_{1}(y)+c_{2} F_{2}(y)\). c Find the expected value and variance of ????. Equation Transcription: Text Transcription: F(y)= 0, if y<0, y2+0.1, if 0</=y<0.5, y, if 0.5</=y<1, 1, if y>/=1. F_1(y) F_2(y) F(y) F(y) c_1F_1(y)+c_2F_2(y)

Problem 162SE A manufacturing plant utilizes 3000 electric light bulbs whose length of life is normally distributed with mean 500 hours and standard deviation 50 hours. To minimize the number of bulbs that burn out during operating hours, all the bulbs are replaced after a given period of operation. How often should the bulbs be replaced if we want not more than 1% of the bulbs to burn out between replacement periods?

Let the density function of a random variable ???? be given by \(f(y)=\left\{\begin{array}{ll} \frac{2}{\pi\left(1+y^{2}\right)}, & -1 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the distribution function. b Find ????(????). Equation Transcription: Text Transcription: f(y)= 2 over pi (1+y^2), -1</=y</=1, 0, elsewhere. E(Y)

Problem 161SE The length of time required to complete a college achievement test is found to be normally distributed with mean 70 minutes and standard deviation 12 minutes. When should the test be terminated if we wish to allow sufficient time for 90% of the students to complete the test?

Consider the nail-firing device of Example 4.15. When the device works, the nail is fired with velocity, ????, with density \(f(v)=\frac{v^{3} e^{-v / 500}}{(500)^{4} \Gamma(4)}\). The device misfires 2% of the time it is used, resulting in a velocity of 0. Find the expected kinetic energy associated with a nail of mass ????. Recall that the kinetic energy, ????, of a mass ???? moving at velocity ???? is \(k=\left(m v^{2}\right) / 2\). The kinetic energy ???? associated with a mass m moving at velocity ???? is given by the expression \(k=\frac{m v^{2}}{2}\). Consider a device that fires a serrated nail into concrete at a mean velocity of 2000 feet per second, where the random velocity ???? possesses a density function given by \(f(v)=\frac{v^{3} e^{-v / 500}}{(500)^{4} \Gamma(4)}, \quad v \geq 0.\) Find the expected kinetic energy associated with a nail of mass ????. \(E(K)=E\left(\frac{m V^{2}}{2}\right)=\frac{m}{2} E\left(V^{2}\right),\) Equation Transcription: Text Transcription: f(v)=v^3e^-v/500 over (500)^4 Gamma(4) k=(mv^2)/2 k=mv^2 over 2 f(v)=v^3e^-v/500 over (500)^4 Gamma(4), v>/=0 E(K)=E(mV^2 over 2)=m over 2 E(V^2)

Problem 163SE Refer to Exercise 4.66. Suppose that five bearings are randomly drawn from production. What is the probability that at least one is defective? Reference A machining operation produces bearings with diameters that are normally distributed with mean 3.0005 inches and standard deviation .0010 inch. Specifications require the bearing diameters to lie in the interval 3.000 ± .0020 inches. Those outside the interval are considered scrap and must be remachined. With the existing machine setting, what fraction of total production will be scrap? a Answer the question, using Table 4, Appendix 3. b Applet Exercise Obtain the answer, using the applet Normal Probabilities.

Problem 164SE The length of life of oil-drilling bits depends upon the types of rock and soil that the drill encounters, but it is estimated that the mean length of life is 75 hours. An oil exploration company purchases drill bits whose length of life is approximately normally distributed with mean 75 hours and standard deviation 12 hours. What proportion of the company’s drill bits a will fail before 60 hours of use? b will last at least 60 hours? c will have to be replaced after more than 90 hours of use?

Let ???? have density function \(f(y)=\left\{\begin{array}{ll} c y e^{-2 y}, & 0 \leq y \leq \infty, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find the value of c that makes \(f(y)\) a density function. b Give the mean and variance for ????. c Give the moment-generating function for ????. Equation Transcription: Text Transcription: f(y)= cye^-2y, 0</=y</=infinity, 0, elsewhere. f(y)

Problem 168E The number of arrivals N at a supermarket checkout counter in the time interval from 0 to t follows a Poisson distribution with mean ?t. Let T denote the length of time until the first arrival. Find the density function for T . [Note: P(T > t0) = P(N = 0 at t = t0).]

Find an expression for \(\mu^{1} k=E\left(Y^{k}\right)\), where the random variable ???? has a beta distribution. Equation Transcription: Text Transcription: mu'k=E(Y^k)

Problem 169SE An argument similar to that of Exercise 4.168 can be used to show that if events are occurring in time according to a Poisson distribution with mean ?t, then the inter arrival times between events have an exponential distribution with mean 1/?. If calls come into a police emergency center at the rate of ten per hour, what is the probability that more than 15 minutes will elapse between the next two calls? Reference The number of arrivals N at a supermarket checkout counter in the time interval from 0 to t follows a Poisson distribution with mean ?t. Let T denote the length of time until the first arrival. Find the density function for T . [Note: P(T > t0) = P(N = 0 at t = t0).]

Use the fact that \(e^{z}=1+z+\frac{z^{2}}{2 !}+\frac{z^{3}}{3 !}+\frac{z^{4}}{4 !}+\cdots\) to expand the moment-generating function of Example 4.16 into a series to find \(\mu_1,\ \mu_2,\ \mu_3,\text{ and }\mu_4\) for the normal random variable. Equation Transcription: Text Transcription: ez=1+z+z^2 over 2!+z^3 over 3!+z^4 over 4!+ cdot cdot cdot mu_1, mu_2, mu_3,and mu_4

Problem 170SE Refer to Exercise 4.168. a If U is the time until the second arrival, show that U has a gamma density function with ? = 2 and ? = 1/?. b Show that the time until the kth arrival has a gamma density with ? = k and ? = 1/?. Reference The number of arrivals N at a supermarket checkout counter in the time interval from 0 to t follows a Poisson distribution with mean ?t. Let T denote the length of time until the first arrival. Find the density function for T . [Note: P(T > t0) = P(N = 0 at t = t0).]

Problem 171SE Suppose that customers arrive at a checkout counter at a rate of two per minute. a What are the mean and variance of thewaiting times between successive customer arrivals? b If a clerk takes three minutes to serve the first customer arriving at the counter, what is the probability that at least one more customer will be waiting when the service to the first customer is completed?

Problem 172SE Calls for dial-in connections to a computer center arrive at an average rate of four per minute. The calls follow a Poisson distribution. If a call arrives at the beginning of a one-minute interval, what is the probability that a second call will not arrive in the next 20 seconds?

Problem 173SE Suppose that plants of a particular species are randomly dispersed over an area so that the number of plants in a given area follows a Poisson distribution with a mean density of ? plants per unit area. If a plant is randomly selected in this area, find the probability density function of the distance to the nearest neighboring plant. [Hint: If R denotes the distance to the nearest neighbor, then P( R > r ) is the same as the probability of seeing no plants in a circle of radius r .]

Problem 174SE The time (in hours) a manager takes to interview a job applicant has an exponential distribution with ? = 1/2. The applicants are scheduled at quarter-hour intervals, beginning at 8:00 A.M., and the applicants arrive exactly on time. When the applicant with an 8:15 A.M. appointment arrives at the manager’s office, what is the probability that he will have to wait before seeing the manager?

Problem 176SE If Y has an exponential distribution with mean ?, find (as a function of ?) the median of Y .

The median value y of a continuous random variable is that value such that \(F(y)=.5\). Find the median value of the random variable in Exercise 4.11. Equation Transcription: Text Transcription: F(y)=.5

Problem 177SE Applet Exercise Use the applet Gamma Probabilities and Quantiles to find the medians of gamma distributed random variables with parameters a ? = 1, ? = 3. Compare your answer with that in Exercise 4.176. b ? = 2, ? = 2. Is the median larger or smaller than E(Y )? c ? = 5, ? = 10. Is the median larger or smaller than E(Y )? d In all of these cases, the median exceeds the mean. How is that reflected in the shapes of the corresponding densities? Reference If Y has an exponential distribution with mean ?, find (as a function of ?) the median of Y .

Problem 178SE Graph the beta probability density function for ? = 3 and ? = 2. a If Y has this beta density function, find P(.1 ? Y ? .2) by using binomial probabilities to evaluate F(y). (See Section 4.7.) b Applet Exercise If Y has this beta density function, find P(.1 ? Y ? .2), using the applet Beta Probabilities and Quantiles. c Applet Exercise If Y has this beta density function, use the applet Beta Probabilities and Quantiles to find the .05 and .95-quantiles for Y . d What is the probability that Y falls between the two quantiles you found in part (c)?

A retail grocer has a daily demand ???? for a certain food sold by the pound, where ???? (measured in hundreds of pounds) has a probability density function given by \(f(y)=\left\{\begin{array}{ll} 3 y^{2}, & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) (She cannot stock over 100 pounds.) The grocer wants to order 100???? pounds of food. She buys the food at 6¢ per pound and sells it at 10¢ per pound. What value of ???? will maximize her expected daily profit? Equation Transcription: Text Transcription: f(y)= 3y^2, 0</=y</=1, 0, elsewhere.

Suppose that ???? has a gamma distribution with \(\alpha=3\) and \(\beta=1\). a Use Poisson probabilities to evaluate \(P(Y \leq 4)\). (See Exercise 4.99.) b Applet Exercise Use the applet Gamma Probabilities and Quantiles to find \(P(Y \leq 4)\). Equation Transcription: Text Transcription: alpha=3 beta=1 P(Y</=4) P(Y</=4)

A random variable ???? is said to have a log-normal distribution if \(X=\ln (Y)\) has a normal distribution. (The symbol ln denotes natural logarithm.) In this case ???? must be nonnegative. The shape of the log-normal probability density function is similar to that of the gamma distribution, with a long tail to the right. The equation of the log-normal density function is given by \(f(y)=\left\{\begin{array}{ll} \frac{1}{\sigma y \sqrt{2 \pi}} e^{-(\ln (y)-\mu)^{2} /\left(2 \sigma^{2}\right)}, & y>0, \\ 0, & \text { elsewhere. } \end{array}\right. \) Because \(ln(y)\) is a monotonic function of y, \(P(Y \leq y)=P[\ln (Y) \leq \ln (y)]=P[X \leq \ln (y)]\), where ???? has a normal distribution with mean \(\mu\) and variance \(\sigma^{2}\). Thus, probabilities for random variables with a log-normal distribution can be found by transforming them into probabilities that can be computed using the ordinary normal distribution. If ???? has a log-normal distribution with \(\mu=4\) and \(\sigma^{2}=1\), find a \(P(Y \leq 4)\). b \(P(Y>8)\). Equation Transcription: Text Transcription: X=ln(Y) f(y)= 1 over sigma y sqrt 2 pi e^-(ln(y)-mu^2/2 sigma^2), y>0, 0, elsewhere. ln(y) P(Y</=y)=P[ln(Y)</=ln(y)]=P[X</=ln(y)] mu sigma^2 mu=4 sigma^2=1 P(Y</=4) P(Y>8)

Let ???? denote a random variable with probability density function given by \(f(y)=(1/2)e^{-|y|},\quad\ \ \ \ -\infty<y<\infty.\) Find the moment-generating function of ???? and use it to find \(E(Y)\). Equation Transcription: Text Transcription: f(y)=(1/2)e^-|y|, -infinity<y<infinity. E(Y)

Let \(f_{1}(y) \text { and } f_{2}(y)\) be density functions and let ???? be a constant such that \(0 \leq a \leq 1\). Consider the function \(f(y)=a f_{1}(y)+(1-a) f_{2}(y)\). a Show that \(f(y)\) is a density function. Such a density function is often referred to as a mixture of two density functions. b Suppose that \(Y_{1}\) is a random variable with density function \(f_{1}(y)\) and that \(E\left(Y_{1}\right)=\mu_{1}\) and \(\operatorname{Var}\left(Y_{1}\right)=\sigma_{1}^{2}\); and similarly suppose that \(Y_{2}\) is a random variable with density function \(f_{2}(y)\) and that \(E\left(Y_{2}\right)=\mu_{2}\) and \(\operatorname{Var}\left(Y_{2}\right)=\sigma_{2}^{2}\). Assume that ???? is a random variable whose density is a mixture of the densities corresponding to \(Y_{1} \text { and } Y_{2}\). Show that i \(E(Y)=a \mu_{1}+(1-\mathrm{a}) \mu_{2}\). ii \(\operatorname{Var}(Y)=a \sigma_{1}^{2}+(1-a) \sigma_{2}^{2}+a(1-a)\left[\mu_{1}-\mu_{2}\right]^{2}\). [Hint: \(E\left(Y_i^2\right)=\mu_i^2+\sigma_i^2,\ i=1,2\).] Equation Transcription: Text Transcription: f_1(y) and f_2(y) 0</=a</=1 f(y)=af_1(y)+(1-a)f_2(y) f(y) Y_1 f_1(y) E(Y_1)=mu_1 Var(Y_1)=sigma_1^2 Y_2 f_2(y) E(Y_2)=mu_2 Var(Y_2)=sigma_2^2 Y_1 and Y_2 E(Y)=amu_1+(1-a)mu_2 Var(Y)=asigma_1^2+(1-a)sigma_2^2+a(1-a)[mu_1-mu_2]^2 E(Y_i^2)=mu_i^2+sigma_i^2,i=1,2

Suppose that ???? is a normally distributed random variable with mean \(\mu\) and variance \(\sigma^{2}\). Use the results of Example 4.16 to find the moment-generating function, mean, and variance of \(Z=\frac{Y-\mu}{\sigma}\). What is the distribution of ????? Why? Equation Transcription: Text Transcription: mu sigma^2 Z=Y-mu over sigma

The random variable ????, with a density function given by \(f(y)=\frac{m y^{m-1}}{\alpha} e^{-y^{m} / \alpha}, \quad 0 \leq y<\infty, \alpha, m>0\) is said to have a Weibull distribution. The Weibull density function provides a good model for the distribution of length of life for many mechanical devices and biological plants and animals. Find the mean and variance for a Weibull distributed random variable with \(m=2\). Equation Transcription: Text Transcription: f(y)=my^m-1 over alpha e^-ym/alpha, 0</=y<infinity, alpha, m>0 m=2

Refer to Exercise 4.186. Resistors used in the construction of an aircraft guidance system have life lengths that follow a Weibull distribution with \(m=2\) and \(\alpha=10\) (with measurements in thousands of hours). a Find the probability that the life length of a randomly selected resistor of this type exceeds 5000 hours. b If three resistors of this type are operating independently, find the probability that exactly one of the three will burn out prior to 5000 hours of use. Equation Transcription: Text Transcription: m=2 =10

Refer to Exercise 4.186. a What is the usual name of the distribution of a random variable that has a Weibull distribution with \(m=1\)? b Derive, in terms of the parameters ? and m, the mean and variance of a Weibull distributed random variable. Equation Transcription: Text Transcription: m=1

If Y has a log-normal distribution with parameters \(\mu\) and \(\sigma^{2}\), it can be shown that \(E(Y)=e^{\left(\mu+\sigma^{2}\right) / 2} \quad \text { and } \quad V(Y)=e^{2 \mu+\sigma^{2}}\left(e^{\sigma^{2}}-1\right)\). The grains composing polycrystalline metals tend to have weights that follow a log-normal distribution. For a type of aluminum, gram weights have a log-normal distribution with \(\mu=3\) and \(\sigma=4\) (in units of \(10^{-2}\) g). a Find the mean and variance of the grain weights. b Find an interval in which at least 75% of the grain weights should lie. [Hint: Use Tchebysheff’s theorem.] c Find the probability that a randomly chosen grain weighs less than the mean grain weight. Equation Transcription: Text Transcription: mu and sigma^2 E(Y)=e(mu+sigma^2)/2 and V(Y)=e^2mu+sigma^2(e^sigma^2-1) mu=3 sigma=4 10^-2

If \(n>2\) is an integer, the distribution with density given by \(f(y)=\left\{\begin{array}{ll} \frac{1}{B(1 / 2,[n-2] / 2)}\left(1-y^{2}\right)^{(n-4) / 2}, & -1 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) is called the ???? distribution. Derive the mean and variance of a random variable with the ???? distribution. Equation Transcription: Text Transcription: n>2 f(y)= 1 over B(1/2, [n-2]/2) (1-y^2)^(n-4)/2, -1</=y</=1, 0, elsewhere.

A function sometimes associated with continuous nonnegative random variables is the failure rate (or hazard rate) function, which is defined by \(r(t)=\frac{f(t)}{1-F(t)}\) for a density function \(f(t)\) with corresponding distribution function \(F(t)\). If we think of the random variable in question as being the length of life of a component, \(r(t)\) is proportional to the probability of failure in a small interval after ????, given that the component has survived up to time ????. Show that, a for an exponential density function, \(r(t)\) is constant. b for a Weibull density function with \(m>1\), \(r(t)\) is an increasing function of ????. (See Exercise 4.186.) Equation Transcription: Text Transcription: r(t)=f(t) over 1-F(t) f(t) F(t) r(t) r(t) m>1 r(t)

Because \(P(Y \leq y \mid Y \geq c)=\frac{F(y)-F(c)}{1-F(c)}\) has the properties of a distribution function, its derivative will have the properties of a probability density function. This derivative is given by \(\frac{f(y)}{1-F(c)}, \quad y \geq c.\) We can thus find the expected value of ????, given that ???? is greater than ????, by using \(E(Y\mid Y\ge c)=\frac{1}{1-F(c)}\int_c^{\infty}yf\ (y)\ dy.\) If ????, the length of life of an electronic component, has an exponential distribution with mean 100 hours, find the expected value of ????, given that this component already has been in use for 50 hours. Equation Transcription: Text Transcription: P(Y</=y|Y>/=c)=F(y)-F(c) over 1-F(c) f(y) over 1-F(c), y>/=c. E(Y|Y>/=c)=1 over 1-F(c) integral c to infinity yf(y)dy.

Suppose that ???? is a continuous random variable with distribution function given by \(F(y)\) and probability density function \(f(y)\). We often are interested in conditional probabilities of the form \(P(Y \leq y \mid Y \geq c)\) for a constant ????. a Show that, for \(y \geq c\), \(P(Y \leq y \mid Y \geq c)=\frac{F(y)-F(c)}{1-F(c)}\). b Show that the function in part (a) has all the properties of a distribution function. c If the length of life ???? for a battery has a Weibull distribution with \(m=2\) and \(\alpha=3\) (with measurements in years), find the probability that the battery will last less than four years, given that it is now two years old. Equation Transcription: Text Transcription: F(y) f(y) P(Y</=y|Y>/=c) y>/=c P(Y<=y|Y>=c)=F(y)-F(c) over 1-F(c) m=2 alpha=3

The velocities of gas particles can be modeled by the Maxwell distribution, whose probability density function is given by \(f(v)=4 \pi\left(\frac{m}{2 \pi K T}\right)^{3 / 2} v^{2} e^{-v^{2}(m /[2 K T])}, \quad v>0,\) where ???? is the mass of the particle, ???? is Boltzmann’s constant, and ???? is the absolute temperature. a Find the mean velocity of these particles. b The kinetic energy of a particle is given by \((1 / 2) m V^{2}\). Find the mean kinetic energy for a particle. Equation Transcription: Text Transcription: f(v)=4pi(m over 2piKT)^3/2 v^2 e^-v^2(m/[2KT]), v>0, (1/2)mV^2

We can show that the normal density function integrates to unity by showing that, if \(u>0\), \(\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-(1 / 2) u y^{2}} d y=\frac{1}{\sqrt{u}}.\) This, in turn, can be shown by considering the product of two such integrals: \(\frac{1}{2\pi}\left(\int_{-\infty}^{\infty}e^{-(1/2)uy^2}dy\right)\left(\int_{-\infty}^{\infty}e^{-(1/2)ux^2}dx\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(1/2)u\left(x^2+y^2\right)}dx\ dy.\) By transforming to polar coordinates, show that the preceding double integral is equal to \(1 / u\). Equation Transcription: Text Transcription: u>0 1 over sqrt 2pi integral -infinity to infinity e^-(1/2)uy^2 dy=1 over sqrt u. 1 over sqrt 2pi (integral -infinity to infinity e^-(1/2)uy^2 dy)(integral -infinity to infinity e-^(1/2)ux^2 dx)=1 over 2pi integral -infinity to infinity integral of -infinity to infinity e-^(1/2)u(x^2+y^2)dx dy. 1/u

Let Z be a standard normal random variable and \(W=\left(Z^{2}+3 Z\right)^{2}\). a Use the moments of ???? (see Exercise 4.199) to derive the mean of ????. b Use the result given in Exercise 4.198 to find a value of ???? such that \(P(W \leq w) \geq .90\). Equation Transcription: Text Transcription: W=(Z^2+3Z)^2 P(W</=w).90

The function \(B(\alpha, \beta)\) is defined by \(B(\alpha, \beta)=\int_{0}^{1} y^{\alpha-1}(1-y)^{\beta-1} d y.\) a Letting \(y=\sin ^{2} \theta\), show that \(B(\alpha,\beta)=2\int_0^{\pi/2}\sin^{2\alpha-1}\theta\cos^{2\beta-1}\theta\ d\theta.\) b Write \(\Gamma(\alpha) \Gamma(\beta)\) as a double integral, transform to polar coordinates, and conclude that \(B(\alpha, \beta)=\frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha+\beta)}.\) Equation Transcription: Text Transcription: B(alpha, beta) B(alpha,beta)=integral 0 to 1 y^alpha-1 (1-y)^beta-1 dy. y=sin^2 theta B(alpha,beta)=2 integral 0 to pi/r sin^2alpha-1 cos^2beta-1 dtheta. Gamma(alpha)Gamma(beta) B(alpha,beta)=Gamma(alpha)Gamma(beta) over Gamma(alpha+beta).

Show that \(\Gamma(1 / 2)=\sqrt{\pi}\) by writing \(\Gamma(1/2)=\int_0^{\infty}y^{-1/2}e^{-y}\ dy\) by making the transformation \(y=(1 / 2) x^{2}\) and by employing the result of Exercise 4.194. Equation Transcription: Text Transcription: (1/2)=sqrt pi (1/2)=integral 0 to infinity y^-1/2 e^-y dy y=(1/2)x^2

Let Z be a standard normal random variable. a Show that the expected values of all odd integer powers of ???? are 0. That is, if \(i=1\), 2, . . . , show that \(E\left(Z^{2 i-1}\right)=0\). [Hint: A function \(\mathrm{g}(\cdot)\) is an odd function if, for all \(y,\ g(-y)=-g(y)\). For any odd function \(g(y),\ \int_{-\infty}^{\infty}g(y)\ dy=0\), if the integral exists.] b If \(i=1\), 2, . . . , show that \(E\left(Z^{2 i}\right)=\frac{2^{i} \Gamma\left(i+\frac{1}{2}\right)}{\sqrt{\pi}}.\) [Hint: A function \(h(\cdot)\) is an even function if, for all \(y,\ h(-y)=h(y)\). For any even function \(h(y),\ \int_{-\infty}^{\infty}h(y)\ dy=2\int_0^{\infty}h(y)\ dy\) if the integrals exist. Use this fact, make the change of variable \(w=z^{2} / 2\), and use what you know about the gamma function.] c Use the results in part (b) and in Exercises 4.81(b) and 4.194 to derive \(E\left(Z^2\right),\ E\left(Z^4\right),\ E\left(Z^6\right),\text{ and }E\left(Z^8\right)\). d If \(i=1\), 2, . . . , show that \(E\left(Z^{2 i}\right)=\prod_{j=1}^{i}(2 j-1).\) This implies that the ith even moment is the product of the first ???? odd integers. Equation Transcription: Text Transcription: i=1 E(Z^2i-1)=0 g(cdot) y,g(-y)=-g(y) g(y), integral -infinity to infinity g(y) dy=0 i=1 E(Z^2i)=2^i Gamma(i+1over 2)over sqrt pi. h(cdot) y,h(-y)=h(y) h(y), integral -infinity to infinity h(y) dy=2 integral 0 to infinity h(y) dy w=z^2/2 E(Z^2), E(Z^4), E(Z^6), and E(Z^8) i=1 E(Z^2i)=product j=1 to i (2j-1).

The Markov Inequality Let \(g(Y)\) be a function of the continuous random variable ????, with \(E(|g(Y)|)<\infty\). Show that, for every positive constant ????, \(P(|g(Y)|\le k)\ge1-\frac{E(|g(Y)|)}{k}.\) [Note: This inequality also holds for discrete random variables, with an obvious adaptation in the proof.] Equation Transcription: Text Transcription: g(Y) E(|g(Y)|)<infinity P(|g(Y)|</=k)>/=1-E(|g(Y)|) over k.

Suppose that a random variable ???? has a probability density function given by \(f(y)=\left\{\begin{array}{ll} 6 y(1-y), & 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. \) a Find \(F(y)\). b Graph \(F(y)\) and \(f(y)\). c Find \(P(.5 \leq Y \leq .8)\). Equation Transcription: Text Transcription: f(y)= 6y(1-y), 0</=y</=1, 0, elsewhere. F(y) f(y) P(.5</=Y</=.8)