 4.1E: Let Y be a random variable with p(y) given in the table below. a Gi...
 4.2E: A box contains five keys, only one of which will open a lock. Keys ...
 4.3E: A Bernoulli random variable is one that assumes only two values, 0 ...
 4.4E: Let Y be a binomial random variable with n = 1 and success probabil...
 4.5E: Suppose that Y is a random variable that takes on only integer valu...
 4.6E: Consider a random variable with a geometric distribution (Section 3...
 4.7E: Let Y be a binomial random variable with n = 10 and p = .2.a Use Ta...
 4.8E: Suppose that Y has density function a Find the value of k that make...
 4.9E: A random variable Y has the following distribution function: a Is Y...
 4.10E: Refer to the density function given in Exercise 4.8.a Find the .95...
 4.11E: Suppose that Y possesses the density function a Find the value of c...
 4.12E: The length of time to failure (in hundreds of hours) for a transist...
 4.13E: A supplier of kerosene has a 150gallon tank that is filled at the ...
 4.14E: A gas station operates two pumps, each of which can pump up to 10,0...
 4.15E: As a measure of intelligence, mice are timed when going through a m...
 4.16E: Let Y possess a density function a Find c.b Find F(y).c Graph f (y)...
 4.17E: The length of time required by students to complete a onehour exam...
 4.18E: Let Y have the density function given by a Find c.b Find F(y).c Gra...
 4.19E: Let the distribution function of a random variable Y be a Find the ...
 4.20E: If, as in Exercise 4.16, Y has density function find the mean and v...
 4.21E: If, as in Exercise 4.17, Y has density function find the mean and v...
 4.22E: If, as in Exercise 4.18, Y has density function find the mean and v...
 4.23E: Prove Theorem 4.5.Reference
 4.24E: If Y is a continuous random variable with density function f (y), u...
 4.25E: If, as in Exercise 4.19, Y has distribution function find the mean ...
 4.26E: If Y is a continuous random variable with mean ? and variance ? 2 a...
 4.27E: For certain ore samples, the proportion Y of impurities per sample ...
 4.28E: The proportion of time per day that all checkout counters in a supe...
 4.29E: The temperature Y at which a thermostatically controlled switch tur...
 4.30E: The proportion of time Y that an industrial robot is in operation d...
 4.31E: Daily total solar radiation for a specified location in Florida in ...
 4.32E: Weekly CPU time used by an accounting firm has probability density ...
 4.33E: The pH of water samples from a specific lake is a random variable Y...
 4.34E: Suppose that Y is a continuous random variable with density f (y) t...
 4.35E: If Y is a continuous random variable such that E[(Y ?a)2] < ? for a...
 4.36E: Is the result obtained in Exercise 4.35 also valid for discrete ran...
 4.37E: If Y is a continuous random variable with density function f (y) th...
 4.38E: Suppose that Y has a uniform distribution over the interval (0, 1)....
 4.39E: If a parachutist lands at a random point on a line between markers ...
 4.40E: Suppose that three parachutists operate independently as described ...
 4.41E: A random variable Y has a uniform distribution over the interval (?...
 4.42E: The median of the distribution of a continuous random variable Y is...
 4.43E: A circle of radius r has area A = ? r 2. If a random circle has a r...
 4.44E: The change in depth of a river from one day to the next, measured (...
 4.45E: Upon studying low bids for shipping contracts, a microcomputer manu...
 4.46E: Refer to Exercise 4.45. Find the expected value of low bids on cont...
 4.47E: The failure of a circuit board interrupts work that utilizes a comp...
 4.48E: If a point is randomly located in an interval (a, b) and if Y denot...
 4.49E: A telephone call arrived at a switchboard at random within a onemi...
 4.50E: Beginning at 12:00 midnight, a computer center is up for one hour a...
 4.51E: The cycle time for trucks hauling concrete to a high way constructi...
 4.52E: Refer to Exercise 4.51. Find the mean and variance of the cycle tim...
 4.53E: The number of defective circuit boards coming off a soldering machi...
 4.54E: In using the triangulation method to determine the range of an acou...
 4.55E: Refer to Exercise 4.54. Suppose that measurement errors are uniform...
 4.56E: Refer to Example 4.7. Find the conditional probability that a custo...
 4.57E: According to Zimmels (1983), the sizes of particles used in sedimen...
 4.58E: Use Table 4, Appendix 3, to find the following probabilities for a ...
 4.59E: If Z is a standard normal random variable, find the value z0 such t...
 4.60E: A normally distributed random variable has density function Using t...
 4.61E: What is the median of a normally distributed random variable with m...
 4.62E: If Z is a standard normal random variable, what isa P ( Z 2 < 1)?b ...
 4.63E: A company that manufactures and bottles apple juice uses a machine ...
 4.64E: The weekly amount of money spent on maintenance and repairs by a co...
 4.65E: In Exercise 4.64, how much should be budgeted for weekly repairs an...
 4.66E: A machining operation produces bearings with diameters that are nor...
 4.67E: In Exercise 4.66, what should the mean diameter be in order that th...
 4.68E: The grade point averages (GPAs) of a large population of college st...
 4.69E: Refer to Exercise 4.68. If students possessing a GPA less than 1.9 ...
 4.70E: Refer to Exercise 4.68. Suppose that three students are randomly se...
 4.71E: Wires manufactured for use in a computer system are specified to ha...
 4.72E: One method of arriving at economic forecasts is to use a consensus ...
 4.73E: The width of bolts of fabric is normally distributed with mean 950 ...
 4.74E: Scores on an examination are assumed to be normally distributed wit...
 4.75E: A softdrink machine can be regulated so that it discharges an aver...
 4.76E: The machine described in Exercise 4.75 has standard deviation ? tha...
 4.77E: The SAT and ACT college entrance exams are taken by thousands of st...
 4.78E: Show that the maximum value of the normal density with parameters ?...
 4.79E: Show that the normal density with parameters ? and ? has inflection...
 4.80E: Assume that Y is normally distributed with mean ? and standard devi...
 4.81E:
 4.82E: Reference
 4.83E: Applet Exercise Use the applet Comparison of Gamma Density Function...
 4.84E: Applet Exercise Refer to Exercise 4.83. Use the applet Comparison o...
 4.85E: Applet Exercise Use the applet Comparison of Gamma Density Function...
 4.86E: Applet Exercise When we discussed the ? 2 distribution in this sect...
 4.87E: Applet Exercise Let Y and W have the distributions given in Exercis...
 4.88E: The magnitude of earthquakes recorded in a region of North America ...
 4.89E: If Y has an exponential distribution and P(Y > 2) = .0821, what isa...
 4.90E: Refer to Exercise 4.88. Of the next ten earthquakes to strike this ...
 4.91E: The operator of a pumping station has observed that demand for wate...
 4.92E: The length of time Y necessary to complete a key operation in the c...
 4.93E: Historical evidence indicates that times between fatal accidents on...
 4.94E: Onehour carbon monoxide concentrations in air samples from a large...
 4.95E: Let Y be an exponentially distributed random variable with mean ?. ...
 4.96E: Suppose that a random variable Y has a probability density function...
 4.97E: A manufacturing plant uses a specific bulk product. The amount of p...
 4.98E: Consider the plant of Exercise 4.97. How much of the bulk product s...
 4.99E: If ? > 0 and ? is a positive integer, the relationship between inco...
 4.100E: Let Y be a gammadistributed random variable where ? is a positive ...
 4.101E: Applet Exercise Refer to Exercise 4.88. Suppose that the magnitude ...
 4.102E: Applet Exercise Refer to Exercise 4.97. Suppose that the amount of ...
 4.103E: Explosive devices used in mining operations produce nearly circular...
 4.104E: The lifetime (in hours) Y of an electronic component is a random va...
 4.105E: Fourweek summer rainfall totals in a section of the Midwest United...
 4.106E: The response times on an online computer terminal have approximatel...
 4.107E: Refer to Exercise 4.106.a Use Tchebysheff’s theorem to give an inte...
 4.108E: Annual incomes for heads of household in a section of a city have a...
 4.109E: The weekly amount of downtime Y (in hours) for an industrial machin...
 4.110E: If Y has a probability density function given by obtain E(Y ) and V...
 4.111E: Suppose that Y has a gamma distribution with parameters ? and ?.a I...
 4.112E: Suppose that Y has a ? 2 distribution with ? degrees of freedom. Us...
 4.113E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.114E: Applet Exercise Refer to Exercise 4.113. Use the applet Comparison ...
 4.115E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.116E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.117E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.118E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.119E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.120E: In Chapter 6 we will see that if Y is beta distributed with paramet...
 4.121E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.122E: Applet Exercise Beta densities with ? < 1 and ? < 1 are difficult t...
 4.123E: The relative humidity Y, when measured at a location, has a probabi...
 4.124E: The percentage of impurities per batch in a chemical product is a r...
 4.125E: Refer to Exercise 4.124. Find the mean and variance of the percenta...
 4.126E: Suppose that a random variable Y has a probability density function...
 4.127E: Verify that if Y has a beta distribution with ? = ? = 1, then Y has...
 4.128E: The weekly repair cost Y for a machine has a probability density fu...
 4.129E: During an eighthour shift, the proportion of time Y that a sheetm...
 4.130E: Prove that the variance of a betadistributed random variable with ...
 4.131E: Errors in measuring the time of arrival of a wave front from an aco...
 4.132E: Proper blending of fine and coarse powders prior to copper sinterin...
 4.133E: The proportion of time per day that all checkout counters in a supe...
 4.134E: In the text of this section, we noted the relationship between the ...
 4.135E: Suppose that Y1 and Y2 are binomial random variables with parameter...
 4.136E: Suppose that the waiting time for the first customer to enter a ret...
 4.137E: Show that the result given in Exercise 3.158 also holds for continu...
 4.138E: Example 4.16 derives the momentgenerating function for Y ? ?, wher...
 4.139E: The momentgenerating function of a normally distributed random var...
 4.140E: Identify the distributions of the random variables with the followi...
 4.141E: If ?1 < ?2, derive the momentgenerating function of a random varia...
 4.142E: Refer to Exercises 4.141 and 4.137. Suppose that Y is uniformly dis...
 4.143E: The momentgenerating function for the gamma random variable is der...
 4.144E: Consider a random variable Y with density function given by a Find ...
 4.145E: A random variable Y has the density function a Find E(e3Y /2).b Fin...
 4.146E: A manufacturer of tires wants to advertise a mileage interval that ...
 4.147E: A machine used to fill cereal boxes dispenses, on the average, ? ou...
 4.148E: Find P(Y ? ? ? 2?) for Exercise 4.16. Compare with the correspond...
 4.149E: Find P(Y ? ? ? 2?) for the uniform random variable. Compare with ...
 4.150E: Find P(Y ? ? ? 2?) for the exponential random variable. Compare w...
 4.151E: Refer to Exercise 4.92. Would you expect C to exceed 2000 very ofte...
 4.152E: Refer to Exercise 4.109. Find an interval that will contain L for a...
 4.153E: Refer to Exercise 4.129. Find an interval for which the probability...
 4.154E: Suppose that Y is a ? 2 distributed random variable with ? = 7 degr...
 4.155E: A builder of houses needs to order some supplies that have a waitin...
 4.156E: The duration Y of longdistance telephone calls (in minutes) monito...
 4.157E: The life length Y of a component used in a complex electronic syste...
 4.158E: Consider the nailfiring device of Example 4.15. When the device wo...
 4.159E: A random variable Y has distribution function
 4.160SE: Let the density function of a random variable Y be given by a Find ...
 4.161SE: The length of time required to complete a college achievement test ...
 4.162SE: A manufacturing plant utilizes 3000 electric light bulbs whose leng...
 4.163SE: Refer to Exercise 4.66. Suppose that five bearings are randomly dra...
 4.164SE: The length of life of oildrilling bits depends upon the types of r...
 4.165SE: Let Y have density function a Find the value of c that makes f (y) ...
 4.166SE: Use the fact that to expand the momentgenerating function of Examp...
 4.167SE: Find an expression for = E(Y k ), where the random variable Y has a...
 4.168SE: The number of arrivals N at a supermarket checkout counter in the t...
 4.169SE: An argument similar to that of Exercise 4.168 can be used to show t...
 4.170SE: Refer to Exercise 4.168.a If U is the time until the second arrival...
 4.171SE: Suppose that customers arrive at a checkout counter at a rate of tw...
 4.172SE: Calls for dialin connections to a computer center arrive at an ave...
 4.173SE: Suppose that plants of a particular species are randomly dispersed ...
 4.174SE: The time (in hours) a manager takes to interview a job applicant ha...
 4.175SE: The median value y of a continuous random variable is that value su...
 4.176SE: If Y has an exponential distribution with mean ?, find (as a functi...
 4.177SE: Applet Exercise Use the applet Gamma Probabilities and Quantiles to...
 4.178SE: Graph the beta probability density function for ? = 3 and ? = 2.a I...
 4.179SE: A retail grocer has a daily demand Y for a certain food sold by the...
 4.180SE: Suppose that Y has a gamma distribution with ? = 3 and ? = 1.a Use ...
 4.181SE: Suppose that Y is a normally distributed random variable with mean ...
 4.182SE: A random variable Y is said to have a lognormal distribution if X ...
 4.183SE: If Y has a lognormal distribution with parameters ? and ? 2, it ca...
 4.184SE: Let Y denote a random variable with probability density function gi...
 4.185SE: Let f1(y) and f2(y) be density functions and let a be a constant su...
 4.186SE: The random variable Y, with a density function given by is said to ...
 4.187SE: Refer to Exercise 4.186. Resistors used in the construction of an a...
 4.188SE: Refer to Exercise 4.186.a What is the usual name of the distributio...
 4.189SE: If n > 2 is an integer, the distribution with density given by is c...
 4.190SE: A function sometimes associated with continuous nonnegative random ...
 4.191SE: Suppose that Y is a continuous random variable with distribution fu...
 4.192SE: The velocities of gas particles can be modeled by the Maxwell distr...
 4.193SE: Because has the properties of a distribution function, its derivati...
 4.194SE: We can show that the normal density function integrates to unity by...
 4.195SE: Let Z be a standard normal random variable and W = (Z 2 + 3Z )2.a U...
 4.196SE: Show that by writing by making the transformation y = (1/2)x2 and b...
 4.197SE: The function B(?, ?) is defined by
 4.198SE: The Markov Inequality Let g(Y ) be a function of the continuous ran...
 4.199SE: Let Z be a standard normal random variable.
 4.200SE: Suppose that Y has a beta distribution with parameters ? and ?.a If...
Solutions for Chapter 4: Mathematical Statistics with Applications 7th Edition
Full solutions for Mathematical Statistics with Applications  7th Edition
ISBN: 9780495110811
Solutions for Chapter 4
Get Full SolutionsThis textbook survival guide was created for the textbook: Mathematical Statistics with Applications , edition: 7. Mathematical Statistics with Applications was written by and is associated to the ISBN: 9780495110811. Since 200 problems in chapter 4 have been answered, more than 96621 students have viewed full stepbystep solutions from this chapter. Chapter 4 includes 200 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Arithmetic mean
The arithmetic mean of a set of numbers x1 , x2 ,…, xn is their sum divided by the number of observations, or ( / )1 1 n xi t n ? = . The arithmetic mean is usually denoted by x , and is often called the average

Bayes’ theorem
An equation for a conditional probability such as PA B (  ) in terms of the reverse conditional probability PB A (  ).

Bias
An effect that systematically distorts a statistical result or estimate, preventing it from representing the true quantity of interest.

Bivariate normal distribution
The joint distribution of two normal random variables

C chart
An attribute control chart that plots the total number of defects per unit in a subgroup. Similar to a defectsperunit or U chart.

Chisquare (or chisquared) random variable
A continuous random variable that results from the sum of squares of independent standard normal random variables. It is a special case of a gamma random variable.

Conidence interval
If it is possible to write a probability statement of the form PL U ( ) ? ? ? ? = ?1 where L and U are functions of only the sample data and ? is a parameter, then the interval between L and U is called a conidence interval (or a 100 1( )% ? ? conidence interval). The interpretation is that a statement that the parameter ? lies in this interval will be true 100 1( )% ? ? of the times that such a statement is made

Consistent estimator
An estimator that converges in probability to the true value of the estimated parameter as the sample size increases.

Continuity correction.
A correction factor used to improve the approximation to binomial probabilities from a normal distribution.

Continuous distribution
A probability distribution for a continuous random variable.

Covariance matrix
A square matrix that contains the variances and covariances among a set of random variables, say, X1 , X X 2 k , , … . The main diagonal elements of the matrix are the variances of the random variables and the offdiagonal elements are the covariances between Xi and Xj . Also called the variancecovariance matrix. When the random variables are standardized to have unit variances, the covariance matrix becomes the correlation matrix.

Cumulative distribution function
For a random variable X, the function of X deined as PX x ( ) ? that is used to specify the probability distribution.

Cumulative normal distribution function
The cumulative distribution of the standard normal distribution, often denoted as ?( ) x and tabulated in Appendix Table II.

Cumulative sum control chart (CUSUM)
A control chart in which the point plotted at time t is the sum of the measured deviations from target for all statistics up to time t

Design matrix
A matrix that provides the tests that are to be conducted in an experiment.

Discrete distribution
A probability distribution for a discrete random variable

Estimate (or point estimate)
The numerical value of a point estimator.

Experiment
A series of tests in which changes are made to the system under study

Exponential random variable
A series of tests in which changes are made to the system under study

Finite population correction factor
A term in the formula for the variance of a hypergeometric random variable.