Computer chips often contain surface imperfections. For a

Determine whether each of the following random variables is discrete or continuous. a. The number of heads in 100 tosses of a coin. b. The length of a rod randomly chosen from a day’s production. c. The final exam score of a randomly chosen student from last semester’s engineering statistics class. d. The age of a randomly chosen Colorado School of Mines student. e. The age that a randomly chosen Colorado School of Mines student will be on his or her next birthday.

Computer chips often contain surface imperfections. For a certain type of computer chip, the probability mass function of the number of defects X is presented in the following table. a. Find \(P(X \leq 2)\). b. Find \(P(X>1)\). c. Find \(\mu_{X}\). d. Find \(\sigma_{X}^{2}\). Equation Transcription: Text Transcription: p(x) P(X{</=}2) P(X>1) mu_X sigma_x^2

A chemical supply company ships a certain solvent in 10-gallon drums. Let X represent the number of drums ordered by a randomly chosen customer. Assume X has the following probability mass function: a. Find the mean number of drums ordered. b. Find the variance of the number of drums ordered. c. Find the standard deviation of the number of drums ordered. d. Let Y be the number of gallons ordered. Find the probability mass function of Y . e. Find the mean number of gallons ordered. f. Find the variance of the number of gallons ordered. g. Find the standard deviation of the number of gallons ordered.

A survey of cars on a certain stretch of highway during morning commute hours showed that 70% had only one occupant, 15% had 2, 10% had 3, 3% had 4, and 2% had 5. Let X represent the number of occupants in a randomly chosen car. a. Find the probability mass function of X. b. Find \(P(X \leq 2)\). c. Find \(P(X>3)\). d. Find \(\mu_{X}\). e. Find \(\sigma_X\). Equation Transcription: Text Transcription: P(X{</=}2) P(X>3) mu_X sigma_X

Let X represent the number of tires with low air pressure on a randomly chosen car. a. Which of the three functions below is a possible probability mass function of X? Explain. b. For the possible probability mass function, compute \(\mu_{X}\) and \(\sigma_{X}^{2}\). Equation Transcription: Text Transcription: p_1(x) p_2(x) p_3(x) mu_X mu_X^2

The repair time (in hours) for a certain machine is a random variable with probability density function \(f(x)=\left\{\begin{array}{cc} x e^{-x} & x>0 \\ 0 & x \leq 0 \end{array}\right. \) a. What is the probability that the repair time is less than 2 hours? b. What is the probability that the repair time is between and 3 hours? c. Find the mean repair time. d. Find the cumulative distribution function of the repair times. Equation Transcription: Text Transcription: f(x)={_0 x{</=}0 ^xe-x x>0

A computer sends a packet of information along a channel and waits for a return signal acknowledging that the packet has been received. If no acknowledgment is received within a certain time, the packet is re-sent. Let represent the number of times the packet is sent. Assume that the probability mass function of is given by \(p(x)= \begin{cases}c x & \text { for } x=1,2,3,4, \text { or } 5 \\ 0 & \text { otherwise }\end{cases}\) where is a constant. a. Find the value of the constant so that \(p(x)\)is a probability mass function. b. Find \(P(X=2)\). c. Find the mean number of times the packet is sent. d. Find the variance of the number of times the packet is sent. e. Find the standard deviation of the number of times the packet is sent. Equation Transcription: Text Transcription: p(x)={_0 otherwise ^cx for x=1,2,3,4, or 5 p(x) P(X=2)

Microprocessing chips are randomly sampled one by one from a large population, and tested to determine if they are acceptable for a certain application. Ninety percent of the chips in the population are acceptable. a. What is the probability that the first chip chosen is acceptable? b. What is the probability that the first chip is unacceptable, and the second is acceptable? c. Let X represent the number of chips that are tested up to and including the first acceptable chip. Find \(P(X=3)\). d. Find the probability mass function of X. Equation Transcription: Text Transcription: P(X=3)

After manufacture, computer disks are tested for errors. Let X be the number of errors detected on a randomly chosen disk. The following table presents values of the cumulative distribution function F(x) of X. a. What is the probability that two or fewer errors are detected? b. What is the probability that more than three errors are detected? c. What is the probability that exactly one error is detected? d. What is the probability that no errors are detected? e. What is the most probable number of errors to be detected?

On 100 different days, a traffic engineer counts the number of cars that pass through a certain intersection between 5 P.M. and 5:05 P.M. The results are presented in the following table. a. Let X be the number of cars passing through the intersection between 5 P.M. and 5:05 P.M. on a randomly chosen day. Someone suggests that for any positive integer x, the probability mass function of X is \(p_1(x)=(0.2)(0.8)^x\). Using this function, compute \(P(X=x)\) for values of x from 0 through 5 inclusive. b. Someone else suggests that for any positive integer x, the probability mass function is \(p_2(x)=(0.4)(0.6)^x\). Using this function, compute \(P(X=x)\) for values of x from 0 through 5 inclusive. c. Compare the results of parts (a) and (b) to the data in the table. Which probability mass function appears to be the better model? Explain. d. Someone says that neither of the functions is a good model since neither one agrees with the data exactly. Is this right? Explain. Equation Transcription: Text Transcription: p_1(x)=(0.2)(0.8)^x P(X=x) p_2(x)=(0.4)(0.6)^x P(X=x)

The element titanium has five stable occurring isotopes, differing from each other in the number of neutrons an atom contains. If is the number of neutrons in a randomly chosen titanium atom, the probability mass function of is given as follows: 24 25 26 27 28 \(p(x)\) a. Find \(\mu_X\). b. Find \(\sigma_X\). Equation Transcription: Text Transcription: p(x) mu_X sigma_X

Refer to Exercise 10. Let Y be the number of chips tested up to and including the second acceptable chip. a. What is the smallest possible value for Y ? b. What is the probability that Y takes on that value? c. Let X represent the number of chips that are tested up to and including the first acceptable chip. Find \(P(Y=3|X=1)\). d. Find \(P(Y=3|X=2)\). e. Find \(P(Y=3)\). Equation Transcription: Text Transcription: P(Y=3|X=1) P(Y=3|X=2) P(Y=3)

Three components are randomly sampled, one at a time, from a large lot. As each component is selected, it is tested. If it passes the test, a success (S) occurs; if it fails the test, a failure (F) occurs. Assume that of the components in the lot will succeed in passing the test. Let represent the number of successes among the three sampled components. a. What are the possible values for b. Find \(P(X=3)\). c. The event that the first component fails and the next two succeed is denoted by FSS. Find (FSS). d. Find and . e. Use the results of parts (c) and (d) to find \(P(X=2)\). f. Find \(P(X=1)\). g. Find \(P(X=0)\). h. Find \(\mu_{X}\). i. Find \(\sigma_{X}^{2}\). j. Let represent the number of successes if four components are sampled. Find \(P(Y=3)\). Equation Transcription: Text Transcription: P(X=3) P(X=2) P(X=1) P(X=0) mu_X sigma_X^2 P(Y=3)

Elongation (in percent) of steel plates treated with aluminum are random with probability density function \(f(x)=\left\{\begin{array}{cl} \frac{x}{250} & 20<x<30 \\ 0 & \text { otherwise } \end{array}\right. \) a. What proportion of steel plates have elongations greater than 25%? b. Find the mean elongation. c. Find the variance of the elongations. d. Find the standard deviation of the elongations. e. Find the cumulative distribution function of the elongations. f. A particular plate elongates 28%. What proportion of plates elongate more than this? Equation Transcription: Text Transcription: f(x)={_0 otherwise^x{over}250 20<x<30

Refer to Exercise 16. A competing process produces rings whose diameters (in centimeters) vary according to the probability density function \(f(x)= \begin{cases}15\left[1-25(x-10.05)^{2}\right] / 4 & \\ & 9.85<x<10.25 \\ 0 & \text { otherwise }\end{cases}\) Specifications call for the diameter to be \(10.0 \pm 0.1 \mathrm{~cm}\). Which process is better, this one or the one in Exercise 16? Explain. Equation Transcription: Text Transcription: f(x)={_0 otherwise ^15[1-25(x-10.05)^2]/4 9.85<x<10.25

A process that manufactures piston rings produces rings whose diameters (in centimeters) vary according to the probability density function \(f(x)= \begin{cases}3\left[1-16(x-10)^{2}\right] & 9.75<x<10.25 \\ 0 & \text { otherwise }\end{cases}\) a. Find the mean diameter of rings manufactured by this process. b. Find the standard deviation of the diameters of rings manufactured by this process. (Hint: Equation 2.36 may be easier to use than Equation 2.37.) c. Find the cumulative distribution function of piston ring diameters. d. What proportion of piston rings have diameters less than 9.75 cm? e. What proportion of piston rings have diameters between 9.75 and 10.25 cm? Equation Transcription: Text Transcription: f(x)={_0 otherwise ^3[1-16(x-10)2] 9.75<x<10.25

The level of impurity (in percent) in the product of a certain chemical process is a random variable with probability density function \(\left\{\begin{array}{cl} \frac{3}{64} x^{2}(4-x) & 0<x<4 \\ 0 & \text { otherwise } \end{array}\right. \) a. What is the probability that the impurity level is greater than 3%? b. What is the probability that the impurity level is between 2% and 3%? c. Find the mean impurity level. d. Find the variance of the impurity levels. e. Find the cumulative distribution function of the impurity level Equation Transcription: Text Transcription: {0 otherwise ^3{over}64 x^2(4-x) 0<x<4

Resistors labeled \(100\ \Omega\) have true resistances that are between \(80\ \Omega\) and \(120\ \Omega\). Let be the mass of a randomly chosen resistor. The probability density function of is given by \(f(x)=\left\{\begin{array}{cl} \frac{x-80}{800} & 80<x<120 \\ 0 & \text { otherwise } \end{array}\right. \) a. What proportion of resistors have resistances less than \(90\ \Omega\)? b. Find the mean resistance. c. Find the standard deviation of the resistances. d. Find the cumulative distribution function of the resistances. Equation Transcription: Text Transcription: 100 ohms 80 ohms 120 ohms f(x)={_0 otherwise ^{x-80}over{800} 80<x<120 90 ohms

The main bearing clearance (in mm) in a certain type of engine is a random variable with probability density function \(f(x)=\left\{\begin{array}{cl} 625 x & 0<x \leq 0.04 \\ 50-625 x & 0.04<x \leq 0.08 \\ 0 & \text { otherwise } \end{array}\right. \) a. What is the probability that the clearance is less than 0.02 mm? b. Find the mean clearance. c. Find the standard deviation of the clearances. d. Find the cumulative distribution function of the clearance. e. Find the median clearance. f. The specification for the clearance is 0.015 to 0.063 mm. What is the probability that the specification is met? Equation Transcription: Text Transcription: f(x)=625 0<x0.04 f(x)=50-625x 0.04<x0.08 f(x)=0 otherwise

The thickness of a washer (in mm) is a random variable with probability density function \(f(x)=\left\{\begin{array}{cl} \frac{3}{52} x(6-x) & 2<x<4 \\ 0 & \text { otherwise } \end{array}\right. \) a. What is the probability that the thickness is less than 2.5 m? b. What is the probability that the thickness is between 2.5 and 3.5 m? c. Find the mean thickness. d. Find the standard deviation \(\sigma\) of the thicknesses. e. Find the probability that the thickness is within \(\pm\sigma\) of the mean. f. Find the cumulative distribution function of the thickness. Equation Transcription: Text Transcription: f(x)={3}over{52}x(6-x) 2<x<4 f(x)=0 otherwise sigma {+/-}sigma

Particles are a major component of air pollution in many areas. It is of interest to study the sizes of contaminating particles. Let X represent the diameter, in micrometers, of a randomly chosen particle. Assume that in a certain area, the probability density function of X is inversely proportional to the volume of the particle; that is, assume that \(f(x)=\left\{\begin{array}{cc} \frac{c}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right. \) where c is a constant. a. Find the value of c so that \(f(x)\) is a probability density function. b. Find the mean particle diameter. c. Find the cumulative distribution function of the particle diameter. d. Find the median particle diameter. e. The term \(\mathrm {PM_{10}}\) refers to particles 10 \(\mu\mathrm m\) or less in diameter. What proportion of the contaminating particles are \(\mathrm {PM_{10}}\)? f. The term \(\mathrm {PM_{2.5}}\) refers to particles 2.5 \(\mu\mathrm m\) or less in diameter. What proportion of the contaminating particles are \(\mathrm {PM_{2.5}}\)? g. What proportion of the \(\mathrm {PM_{10}}\) particles are \(\mathrm {PM_{2.5}}\)? Equation Transcription: Text Transcription: f(x)={_0 x<1 ^{c}over{x3} x{>/=}1 f(x) PM_10 {mu}m PM_10 PM_2.5 {mu}m PM_2.5 PM_10 PM_2.5

The diameter of a rivet (in ) is a random variable with probability density function \(f(x)=\left\{\begin{array}{cc} 6(x-12)(13-x) & 12<x \leq 13 \\ 0 & \text { otherwise } \end{array}\right. \) a. What is the probability that the diameter is less than ? b. Find the mean diameter. c. Find the standard deviation of the diameters. d. Find the cumulative distribution function of the diameter. e. The specification for the diameter is to . What is the probability that the specification is met? Equation Transcription: Text Transcription: f(x)={_0 otherwise ^6(x-12)(13-x) 12<x{</=}13