 5.5.1: Contracts for two construction jobs are randomly assigned to one or...
 5.5.2: Three balanced coins are tossed independently. One of the variables...
 5.5.3: Of nine executives in a business firm, four are married, three have...
 5.5.4: Given here is the joint probability function associated with data o...
 5.5.5: Refer to Example 5.4. The joint density of Y1, the proportion of th...
 5.5.6: Refer to Example 5.3. If a radioactive particle is randomly located...
 5.5.7: Let Y1 and Y2 have joint density function f (y1, y2) = e(y1+y2) , y...
 5.5.8: Let Y1 and Y2 have the joint probability density function given by ...
 5.5.9: Let Y1 and Y2 have the joint probability density function given by ...
 5.5.11: Suppose that Y1 and Y2 are uniformly distributed over the triangle ...
 5.5.12: Let Y1 and Y2 denote the proportions of two different types of comp...
 5.5.13: The joint density function of Y1 and Y2 is given by f (y1, y2) = $ ...
 5.5.14: Suppose that the random variables Y1 and Y2 have joint probability ...
 5.5.15: The management at a fastfood outlet is interested in the joint beh...
 5.5.16: Let Y1 and Y2 denote the proportions of time (out of one workday) d...
 5.5.17: Let (Y1, Y2) denote the coordinates of a point chosen at random ins...
 5.5.18: An electronic system has one each of two different types of compone...
 5.5.19: In Exercise 5.1, we determined that the joint distribution of Y1, t...
 5.5.21: In Exercise 5.3, we determined that the joint probability distribut...
 5.5.22: In Exercise 5.3, we determined that the joint probability distribut...
 5.5.23: In Example 5.4 and Exercise 5.5, we considered the joint density of...
 5.5.24: In Exercise 5.6, we assumed that if a radioactive particle is rando...
 5.5.25: Let Y1 and Y2 have joint density function first encountered in Exer...
 5.5.26: In Exercise 5.8, we derived the fact that f (y1, y2) = 4y1 y2, 0 y1...
 5.5.27: In Exercise 5.9, we determined that f (y1, y2) = 6(1 y2), 0 y1 y2 1...
 5.5.28: In Exercise 5.10, we proved that f (y1, y2) = 1, 0 y1 2, 0 y2 1, 2y...
 5.5.29: Refer to Exercise 5.11. Find a the marginal density functions for Y...
 5.5.31: In Exercise 5.13, the joint density function of Y1 and Y2 is given ...
 5.5.32: Suppose that the random variables Y1 and Y2 have joint probability ...
 5.5.33: Suppose that Y1 is the total time between a customers arrival in th...
 5.5.34: If Y1 is uniformly distributed on the interval (0, 1) and, for 0 < ...
 5.5.35: Refer to Exercise 5.33. If two minutes elapse between a customers a...
 5.5.36: In Exercise 5.16, Y1 and Y2 denoted the proportions of time during ...
 5.5.37: In Exercise 5.18, Y1 and Y2 denoted the lengths of life, in hundred...
 5.5.38: Let Y1 denote the weight (in tons) of a bulk item stocked by a supp...
 5.5.39: Suppose that Y1 and Y2 are independent Poisson distributed random v...
 5.5.41: A quality control plan calls for randomly selecting three items fro...
 5.5.42: The number of defects per yard Y for a certain fabric is known to h...
 5.5.43: Let Y1 and Y2 have joint density function f (y1, y2) and marginal d...
 5.5.44: Prove that the results in Exercise 5.43 also hold for discrete rand...
 5.5.45: In Exercise 5.1, we determined that the joint distribution of Y1, t...
 5.5.46: Refer to Exercise 5.2. The number of heads in three coin tosses is ...
 5.5.47: In Exercise 5.3, we determined that the joint probability distribut...
 5.5.48: In Exercise 5.4, you were given the following joint probability fun...
 5.5.49: In Example 5.4 and Exercise 5.5, we considered the joint density of...
 5.5.51: In Exercise 5.6, we assumed that if a radioactive particle is rando...
 5.5.52: In Exercise 5.8, we derived the fact that f (y1, y2) = 4y1 y2, 0 y1...
 5.5.53: In Exercise 5.9, we determined that f (y1, y2) = 6(1 y2), 0 y1 y2 1...
 5.5.54: In Exercise 5.10, we proved that f (y1, y2) = 1, 0 y1 2, 0 y2 1, 2y...
 5.5.55: Suppose that, as in Exercise 5.11, Y1 and Y2 are uniformly distribu...
 5.5.56: In Exercise 5.12, we were given the following joint probability den...
 5.5.57: In Exercises 5.13 and 5.31, the joint density function of Y1 and Y2...
 5.5.58: Suppose that the random variables Y1 and Y2 have joint probability ...
 5.5.59: If Y1 is the total time between a customers arrival in the store an...
 5.5.61: In Exercise 5.18, Y1 and Y2 denoted the lengths of life, in hundred...
 5.5.62: Suppose that the probability that a head appears when a coin is tos...
 5.5.63: Let Y1 and Y2 be independent exponentially distributed random varia...
 5.5.64: Let Y1 and Y2 be independent random variables that are both uniform...
 5.5.65: Suppose that, for 1 1, the probability density function of (Y1, Y2)...
 5.5.66: Let F1(y1) and F2(y2) be two distribution functions. For any , 1 1,...
 5.5.67: In Section 5.2, we argued that if Y1 and Y2 have joint cumulative d...
 5.5.68: A bus arrives at a bus stop at a uniformly distributed time over th...
 5.5.69: The length of life Y for fuses of a certain type is modeled by the ...
 5.5.71: Two telephone calls come into a switchboard at random times in a fi...
 5.5.72: In Exercise 5.1, we determined that the joint distribution of Y1, t...
 5.5.73: In Exercise 5.3, we determined that the joint probability distribut...
 5.5.74: Refer to Exercises 5.6, 5.24, and 5.50. Suppose that a radioactive ...
 5.5.75: Refer to Exercises 5.6, 5.24, and 5.50. Suppose that a radioactive ...
 5.5.76: In Exercise 5.8, we derived the fact that f (y1, y2) = $ 4y1 y2, 0 ...
 5.5.77: In Exercise 5.9, we determined that f (y1, y2) = $ 6(1 y2), 0 y1 y2...
 5.5.78: In Exercise 5.10, we proved that f (y1, y2) = $ 1, 0 y1 2, 0 y2 1, ...
 5.5.79: Suppose that, as in Exercise 5.11, Y1 and Y2 are uniformly distribu...
 5.5.81: In Exercise 5.18, Y1 and Y2 denoted the lengths of life, in hundred...
 5.5.82: In Exercise 5.38, we determined that the joint density function for...
 5.5.83: In Exercise 5.42, we determined that the unconditional probability ...
 5.5.84: In Exercise 5.62, we considered two individuals who each tossed a c...
 5.5.85: In Exercise 5.65, we considered random variables Y1 and Y2 that, fo...
 5.5.86: Suppose that Z is a standard normal random variable and that Y1 and...
 5.5.87: Suppose that Y1 and Y2 are independent 2 random variables with 1 an...
 5.5.88: Suppose that you are told to toss a die until you have observed eac...
 5.5.89: In Exercise 5.1, we determined that the joint distribution of Y1, t...
 5.5.91: In Exercise 5.3, we determined that the joint probability distribut...
 5.5.92: In Exercise 5.9, we determined that f (y1, y2) = 6(1 y2), 0 y1 y2 1...
 5.5.93: Let the discrete random variables Y1 and Y2 have the joint probabil...
 5.5.94: Let Y1 and Y2 be uncorrelated random variables and consider U1 = Y1...
 5.5.95: Suppose that, as in Exercises 5.11 and 5.79, Y1 and Y2 are uniforml...
 5.5.96: Suppose that the random variables Y1 and Y2 have means 1 and 2 and ...
 5.5.97: The random variables Y1 and Y2 are such that E(Y1) = 4, E(Y2) = 1, ...
 5.5.98: How big or small can Cov(Y1, Y2) be? Use the fact that 2 1 to show ...
 5.5.99: If c is any constant and Y is a random variable such that E(Y ) exi...
 5.5.101: In Exercise 5.65, we considered random variables Y1 and Y2 that, fo...
 5.5.102: A firm purchases two types of industrial chemicals. Type I chemical...
 5.5.103: Assume that Y1, Y2, and Y3 are random variables, with E(Y1) = 2, E(...
 5.5.104: In Exercise 5.3, we determined that the joint probability distribut...
 5.5.105: In Exercise 5.8, we established that f (y1, y2) = 4y1 y2, 0 y1 1, 0...
 5.5.106: In Exercise 5.9, we determined that f (y1, y2) = 6(1 y2), 0 y1 y2 1...
 5.5.107: In Exercise 5.12, we were given the following joint probability den...
 5.5.108: If Y1 is the total time between a customers arrival in the store an...
 5.5.109: In Exercise 5.16, Y1 and Y2 denoted the proportions of time that em...
 5.5.111: A retail grocery merchant figures that her daily gain X from sales ...
 5.5.112: In Exercise 5.18, Y1 and Y2 denoted the lengths of life, in hundred...
 5.5.113: Suppose that Y1 and Y2 have correlation coefficient Y1,Y2 and for c...
 5.5.114: For the daily output of an industrial operation, let Y1 denote the ...
 5.5.115: Refer to Exercise 5.88. If Y denotes the number of tosses of the di...
 5.5.116: Refer to Exercise 5.75. Use Theorem 5.12 to explain why V(Y1 + Y2) ...
 5.5.117: A population of N alligators is to be sampled in order to obtain an...
 5.5.118: The total sustained load on the concrete footing of a planned build...
 5.5.119: A learning experiment requires a rat to run a maze (a network of pa...
 5.5.121: Refer to Exercise 5.117. Suppose that the number N of alligators in...
 5.5.122: The weights of a population of mice fed on a certain diet since bir...
 5.5.123: The National Fire Incident Reporting Service stated that, among res...
 5.5.124: The typical cost of damages caused by a fire in a family home is $2...
 5.5.125: When commercial aircraft are inspected, wing cracks are reported as...
 5.5.126: A large lot of manufactured items contains 10% with exactly one def...
 5.5.127: Refer to Exercise 5.126. Let Y denote the number of items among the...
 5.5.128: Let Y1 and Y2 have a bivariate normal distribution. a Show that the...
 5.5.129: Let Y1 and Y2 have a bivariate normal distribution. Show that the c...
 5.5.131: Let Y1 and Y2 be independent normally distributed random variables ...
 5.5.132: Refer to Exercise 5.131. What are the marginal distributions of U1 ...
 5.5.133: In Exercise 5.9, we determined that f (y1, y2) = 6(1 y2), 0 y1 y2 1...
 5.5.134: In Examples 5.32 and 5.33, we determined that if Y is the number of...
 5.5.135: In Exercise 5.41, we considered a quality control plan that calls f...
 5.5.136: In Exercise 5.42, the number of defects per yard in a certain fabri...
 5.5.137: In Exercise 5.38, we assumed that Y1, the weight of a bulk item sto...
 5.5.138: Assume that Y denotes the number of bacteria per cubic centimeter i...
 5.5.139: Suppose that a company has determined that the the number of jobs p...
 5.5.141: Let Y1 have an exponential distribution with mean and the condition...
 5.5.142: Suppose that Y has a binomial distribution with parameters n and p ...
 5.5.143: If Y1 and Y2 are independent random variables, each having a normal...
 5.5.144: Prove Theorem 5.9 when Y1 and Y2 are independent discrete random va...
 5.5.145: A technician starts a job at a time Y1 that is uniformly distribute...
 5.5.146: A target for a bomb is in the center of a circle with radius of 1 m...
 5.5.147: Two friends are to meet at the library. Each independently and rand...
 5.5.148: A committee of three people is to be randomly selected from a group...
 5.5.149: Let Y1 and Y2 have a joint density function given by f (y1, y2) = 3...
 5.5.151: The lengths of life Y for a type of fuse has an exponential distrib...
 5.5.152: In the production of a certain type of copper, two types of copper ...
 5.5.153: Suppose that the number of eggs laid by a certain insect has a Pois...
 5.5.154: In a clinical study of a new drug formulated to reduce the effects ...
 5.5.155: Suppose that Y1, Y2, and Y3 are independent 2distributed random va...
 5.5.156: Refer to Exercise 5.86. Suppose that Z is a standard normal random ...
 5.5.157: A forester studying diseased pine trees models the number of diseas...
 5.5.158: A coin has probability p of coming up heads when tossed. In n indep...
 5.5.159: The negative binomial random variable Y was defined in Section 3.6 ...
 5.5.161: Suppose that we are to observe two independent random samples: Y1, ...
 5.5.162: In Exercise 5.65, you determined that, for 1 1, the probability den...
 5.5.163: Refer to Exercise 5.66. If F1(y1) and F2(y2) are two distribution f...
 5.5.164: Let X1, X2, and X3 be random variables, either continuous or discre...
 5.5.165: Let X1, X2, and X3 have a multinomial distribution with probability...
 5.5.166: A box contains N1 white balls, N2 black balls, and N3 red balls (N1...
 5.5.167: Let Y1 and Y2 be jointly distributed random variables with finite v...
Solutions for Chapter 5: Multivariate Probability Distributions
Full solutions for Mathematical Statistics with Applications  7th Edition
ISBN: 9780495110811
Solutions for Chapter 5: Multivariate Probability Distributions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 5: Multivariate Probability Distributions includes 151 full stepbystep solutions. Mathematical Statistics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495110811. This textbook survival guide was created for the textbook: Mathematical Statistics with Applications , edition: 7th. Since 151 problems in chapter 5: Multivariate Probability Distributions have been answered, more than 57255 students have viewed full stepbystep solutions from this chapter.

2 k p  factorial experiment
A fractional factorial experiment with k factors tested in a 2 ? p fraction with all factors tested at only two levels (settings) each

Alias
In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.

Bayes’ estimator
An estimator for a parameter obtained from a Bayesian method that uses a prior distribution for the parameter along with the conditional distribution of the data given the parameter to obtain the posterior distribution of the parameter. The estimator is obtained from the posterior distribution.

Bernoulli trials
Sequences of independent trials with only two outcomes, generally called “success” and “failure,” in which the probability of success remains constant.

Bivariate distribution
The joint probability distribution of two random variables.

Causeandeffect diagram
A chart used to organize the various potential causes of a problem. Also called a ishbone diagram.

Central tendency
The tendency of data to cluster around some value. Central tendency is usually expressed by a measure of location such as the mean, median, or mode.

Chisquare test
Any test of signiicance based on the chisquare distribution. The most common chisquare tests are (1) testing hypotheses about the variance or standard deviation of a normal distribution and (2) testing goodness of it of a theoretical distribution to sample data

Confounding
When a factorial experiment is run in blocks and the blocks are too small to contain a complete replicate of the experiment, one can run a fraction of the replicate in each block, but this results in losing information on some effects. These effects are linked with or confounded with the blocks. In general, when two factors are varied such that their individual effects cannot be determined separately, their effects are said to be confounded.

Continuous distribution
A probability distribution for a continuous random variable.

Contour plot
A twodimensional graphic used for a bivariate probability density function that displays curves for which the probability density function is constant.

Control chart
A graphical display used to monitor a process. It usually consists of a horizontal center line corresponding to the incontrol value of the parameter that is being monitored and lower and upper control limits. The control limits are determined by statistical criteria and are not arbitrary, nor are they related to speciication limits. If sample points fall within the control limits, the process is said to be incontrol, or free from assignable causes. Points beyond the control limits indicate an outofcontrol process; that is, assignable causes are likely present. This signals the need to ind and remove the assignable causes.

Cook’s distance
In regression, Cook’s distance is a measure of the inluence of each individual observation on the estimates of the regression model parameters. It expresses the distance that the vector of model parameter estimates with the ith observation removed lies from the vector of model parameter estimates based on all observations. Large values of Cook’s distance indicate that the observation is inluential.

Density function
Another name for a probability density function

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

Discrete uniform random variable
A discrete random variable with a inite range and constant probability mass function.

Error variance
The variance of an error term or component in a model.

Firstorder model
A model that contains only irstorder terms. For example, the irstorder response surface model in two variables is y xx = + ?? ? ? 0 11 2 2 + + . A irstorder model is also called a main effects model

Geometric mean.
The geometric mean of a set of n positive data values is the nth root of the product of the data values; that is, g x i n i n = ( ) = / w 1 1 .

Hat matrix.
In multiple regression, the matrix H XXX X = ( ) ? ? 1 . This a projection matrix that maps the vector of observed response values into a vector of itted values by yˆ = = X X X X y Hy ( ) ? ? ?1 .
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