Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer
Publisher: Cengage Learning
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Chapter 6.5, Problem 57E
To determine
Prove that
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Chapter 6 Solutions
Mathematical Statistics with Applications
Ch. 6.3 - Let Y be a random variable with probability...Ch. 6.3 - Prob. 2ECh. 6.3 - Prob. 3ECh. 6.3 - The amount of flour used per day by a bakery is a...Ch. 6.3 - Prob. 5ECh. 6.3 - The joint distribution of amount of pollutant...Ch. 6.3 - Suppose that Z has a standard normal distribution....Ch. 6.3 - Assume that Y has a beta distribution with...Ch. 6.3 - Prob. 9ECh. 6.3 - The total time from arrival to completion of...
Ch. 6.3 - Suppose that two electronic components in the...Ch. 6.3 - Prob. 12ECh. 6.3 - If Y1 and Y2 are independent exponential random...Ch. 6.3 - Prob. 14ECh. 6.3 - Prob. 15ECh. 6.3 - Prob. 16ECh. 6.3 - Prob. 17ECh. 6.3 - A member of the Pareto family of distributions...Ch. 6.3 - Prob. 19ECh. 6.3 - Let the random variable Y possess a uniform...Ch. 6.3 - Prob. 21ECh. 6.4 - Prob. 23ECh. 6.4 - In Exercise 6.4, we considered a random variable Y...Ch. 6.4 - Prob. 25ECh. 6.4 - Prob. 26ECh. 6.4 - Prob. 27ECh. 6.4 - Let Y have a uniform (0, 1) distribution. Show...Ch. 6.4 - Prob. 29ECh. 6.4 - A fluctuating electric current I may be considered...Ch. 6.4 - The joint distribution for the length of life of...Ch. 6.4 - Prob. 32ECh. 6.4 - The proportion of impurities in certain ore...Ch. 6.4 - A density function sometimes used by engineers to...Ch. 6.4 - Prob. 35ECh. 6.4 - Refer to Exercise 6.34. Let Y1 and Y2 be...Ch. 6.5 - Let Y1, Y2,, Yn be independent and identically...Ch. 6.5 - Let Y1 and Y2 be independent random variables with...Ch. 6.5 - Prob. 39ECh. 6.5 - Prob. 40ECh. 6.5 - Prob. 41ECh. 6.5 - A type of elevator has a maximum weight capacity...Ch. 6.5 - Prob. 43ECh. 6.5 - Prob. 44ECh. 6.5 - The manager of a construction job needs to figure...Ch. 6.5 - Suppose that Y has a gamma distribution with =...Ch. 6.5 - A random variable Y has a gamma distribution with ...Ch. 6.5 - Prob. 48ECh. 6.5 - Let Y1 be a binomial random variable with n1...Ch. 6.5 - Let Y be a binomial random variable with n trials...Ch. 6.5 - Prob. 51ECh. 6.5 - Prob. 52ECh. 6.5 - Let Y1,Y2,,Yn be independent binomial random...Ch. 6.5 - Prob. 54ECh. 6.5 - Customers arrive at a department store checkout...Ch. 6.5 - The length of time necessary to tune up a car is...Ch. 6.5 - Prob. 57ECh. 6.5 - Prob. 58ECh. 6.5 - Prob. 59ECh. 6.5 - Prob. 60ECh. 6.5 - Prob. 61ECh. 6.5 - Prob. 62ECh. 6.6 - In Example 6.14, Y1 and Y2 were independent...Ch. 6.6 - Refer to Exercise 6.63 and Example 6.14. Suppose...Ch. 6.6 - Prob. 65ECh. 6.6 - Prob. 66ECh. 6.6 - Prob. 67ECh. 6.6 - Prob. 68ECh. 6.6 - Prob. 71ECh. 6 - Let Y1 and Y2 be independent and uniformly...Ch. 6 - As in Exercise 6.72, let Y1 and Y2 be independent...Ch. 6 - Let Y1, Y2,, Yn be independent, uniformly...Ch. 6 - Prob. 75SECh. 6 - Prob. 76SECh. 6 - Prob. 77SECh. 6 - Prob. 78SECh. 6 - Refer to Exercise 6.77. If Y1,Y2,,Yn are...Ch. 6 - Prob. 80SECh. 6 - Let Y1, Y2,, Yn be independent, exponentially...Ch. 6 - Prob. 82SECh. 6 - Prob. 83SECh. 6 - Prob. 84SECh. 6 - Let Y1 and Y2 be independent and uniformly...Ch. 6 - Prob. 86SECh. 6 - Prob. 87SECh. 6 - Prob. 88SECh. 6 - Let Y1, Y2, . . . , Yn denote a random sample from...Ch. 6 - Prob. 90SECh. 6 - Prob. 91SECh. 6 - Prob. 92SECh. 6 - Prob. 93SECh. 6 - Prob. 94SECh. 6 - Prob. 96SECh. 6 - Prob. 97SECh. 6 - Prob. 98SECh. 6 - Prob. 99SECh. 6 - The time until failure of an electronic device has...Ch. 6 - Prob. 101SECh. 6 - Prob. 103SECh. 6 - Prob. 104SECh. 6 - Prob. 105SECh. 6 - Prob. 106SECh. 6 - Prob. 107SECh. 6 - Prob. 108SECh. 6 - Prob. 109SECh. 6 - Prob. 110SECh. 6 - Prob. 111SECh. 6 - Prob. 112SECh. 6 - Prob. 113SECh. 6 - Prob. 114SECh. 6 - Prob. 115SECh. 6 - Prob. 116SE
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- Let X1 and X2 be independent chi-squared random variables with r1 and r2 degrees of freedom, respectively. Show that, (a) U = X1/(X1+X2) has a beta distribution with alpha = r1/2 and beta = r2/2. (b) V = X2/(X1+X2) has a beta distribution with alpha = r2/2 and beta = r1/2arrow_forwardX is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2arrow_forwardLet X and Y be two random variables with E (X) = 1, E (Y) = 2, Var (X) = 1, Var (Y) = 2, !! Cov (X, Y) = 0.5. For what values of a and b such that the random variable aX + bY have mean 3 and variance 4 ?arrow_forward
- X1 and X2 are independent random variables such that Xi has PDF fXi(x)={λiexp(−λix) when x≥0, 0 otherwise}. What is P[X2<X1]?arrow_forwardLet Xi and Yi be random variables with Var(Xi) = σx2 and Var(Yi) = σy2 for all i ∈ {1, . . . , n}. Assume that each pair (Xi, Yi) has correlation Corr(Xi, Yi) = ρ, but that (Xi,Yi) and (Xj,Yj) are independent for all i ̸= j. (a) What is Cov(Xi,Yi) in terms of σx, σy and ρ? (b) Show that Cov(Xi,Y ̄) = (ρσxσy)/n, where Y ̄ is the average of the Yi (c) Determine Cov(X ̄,Y ̄). B2. Consider the random variables Xi and Yi from question B1 again. (a) Show that the sample covariance is an unbiased estimator of Cov(X1,Y1). Hint: consider the equality Xi − X ̄ = (Xi − μ) − (X ̄ − μ). (b) Can you conclude from the statement in part (a) that the sample correlation is an unbiased estimator of Corr(X1, Y1)? Justify your answer.arrow_forwardLet Y1 and Y2 be independent random variables that are both uniformly distributed on the interval (0, 1). Find P( Y1 < 2Y2 | Y1 < 3Y2).arrow_forward
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