Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer
Publisher: Cengage Learning
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Chapter 5, Problem 155SE
a.
To determine
Calculate the value of
b.
To determine
Explain the reason that the answer of Part (a) is positive.
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4.63
Three zero mean, unit variance random variables X, Y, and Z are added
to form a new random variable, W = X + Y + Z. Random variables X
and Y are uncorrelated, X and Z have a correlation coefficient of 1/2, and
Y and Z have a correlation coefficient of - 1/2.
a) Find the variance of W.
b) Find the correlation coefficient between W and X.
c) Find the correlation coefficient between W and the sum of Y and Z.
Let X, and X, be independent random variables with mean u and variance o?.
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2
Chapter 5 Solutions
Mathematical Statistics with Applications
Ch. 5.2 - Contracts for two construction jobs are randomly...Ch. 5.2 - Three balanced coins are tossed independently. One...Ch. 5.2 - Of nine executives in a business firm, four are...Ch. 5.2 - Given here is the joint probability function...Ch. 5.2 - Refer to Example 5.4. The joint density of Y1, the...Ch. 5.2 - Prob. 6ECh. 5.2 - Let Y1 and Y2 have joint density function...Ch. 5.2 - Prob. 8ECh. 5.2 - Prob. 9ECh. 5.2 - An environmental engineer measures the amount (by...
Ch. 5.2 - Suppose that Y1 and Y2 are uniformly distributed...Ch. 5.2 - Prob. 12ECh. 5.2 - Prob. 13ECh. 5.2 - Prob. 14ECh. 5.2 - The management at a fast-food outlet is interested...Ch. 5.2 - Let Y1 and Y2 denote the proportions of time (out...Ch. 5.2 - Let (Y1, Y2) denote the coordinates of a point...Ch. 5.2 - Prob. 18ECh. 5.3 - In Exercise 5.1, we determined that the joint...Ch. 5.3 - Refer to Exercise 5.2. a Derive the marginal...Ch. 5.3 - In Exercise 5.3, we determined that the joint...Ch. 5.3 - In Exercise 5.4, you were given the following...Ch. 5.3 - In Example 5.4 and Exercise 5.5, we considered the...Ch. 5.3 - Prob. 24ECh. 5.3 - Prob. 25ECh. 5.3 - Prob. 26ECh. 5.3 - Prob. 27ECh. 5.3 - In Exercise 5.10, we proved that...Ch. 5.3 - Prob. 29ECh. 5.3 - In Exercise 5.12, we were given the following...Ch. 5.3 - In Exercise 5.13, the joint density function of Y1...Ch. 5.3 - Prob. 32ECh. 5.3 - Suppose that Y1 is the total time between a...Ch. 5.3 - Prob. 34ECh. 5.3 - Prob. 35ECh. 5.3 - Prob. 36ECh. 5.3 - Prob. 37ECh. 5.3 - Let Y1 denote the weight (in tons) of a bulk item...Ch. 5.3 - Prob. 39ECh. 5.3 - Prob. 40ECh. 5.3 - Prob. 41ECh. 5.3 - Prob. 42ECh. 5.4 - Let Y1 and Y2 have joint density function f(y1,...Ch. 5.4 - Prob. 44ECh. 5.4 - Prob. 45ECh. 5.4 - Prob. 46ECh. 5.4 - In Exercise 5.3, we determined that the joint...Ch. 5.4 - In Exercise 5.4, you were given the following...Ch. 5.4 - In Example 5.4 and Exercise 5.5, we considered the...Ch. 5.4 - Prob. 50ECh. 5.4 - Prob. 51ECh. 5.4 - Prob. 52ECh. 5.4 - Prob. 53ECh. 5.4 - Prob. 54ECh. 5.4 - Prob. 55ECh. 5.4 - In Exercise 5.12, we were given the following...Ch. 5.4 - Prob. 57ECh. 5.4 - Suppose that the random variables Y1 and Y2 have...Ch. 5.4 - If Y1 is the total time between a customers...Ch. 5.4 - Prob. 60ECh. 5.4 - Prob. 61ECh. 5.4 - Prob. 62ECh. 5.4 - Let Y1 and Y2 be independent exponentially...Ch. 5.4 - Prob. 64ECh. 5.4 - Prob. 65ECh. 5.4 - Let F1(y1) and F2(y2) be two distribution...Ch. 5.4 - Prob. 67ECh. 5.4 - Prob. 68ECh. 5.4 - The length of life Y for fuses of a certain type...Ch. 5.4 - A bus arrives at a bus stop at a uniformly...Ch. 5.4 - Prob. 71ECh. 5.6 - In Exercise 5.1, we determined that the joint...Ch. 5.6 - Prob. 73ECh. 5.6 - Refer to Exercises 5.6, 5.24, and 5.50. Suppose...Ch. 5.6 - Prob. 75ECh. 5.6 - Prob. 76ECh. 5.6 - Prob. 77ECh. 5.6 - Prob. 78ECh. 5.6 - Suppose that, as in Exercise 5.11, Y1 and Y2 are...Ch. 5.6 - In Exercise 5.16, Y1 and Y2 denoted the...Ch. 5.6 - In Exercise 5.18, Y1 and Y2 denoted the lengths of...Ch. 5.6 - In Exercise 5.38, we determined that the joint...Ch. 5.6 - Prob. 83ECh. 5.6 - In Exercise 5.62, we considered two individuals...Ch. 5.6 - Prob. 85ECh. 5.6 - Prob. 86ECh. 5.6 - Prob. 87ECh. 5.6 - Prob. 88ECh. 5.7 - In Exercise 5.1, we determined that the joint...Ch. 5.7 - Prob. 90ECh. 5.7 - In Exercise 5.8, we derived the fact that...Ch. 5.7 - Prob. 92ECh. 5.7 - Suppose that, as in Exercises 5.11 and 5.79, Y1...Ch. 5.7 - Prob. 94ECh. 5.7 - Prob. 95ECh. 5.7 - Prob. 96ECh. 5.7 - The random variables Y1 and Y2 are such that E(Y1)...Ch. 5.7 - Prob. 98ECh. 5.7 - Prob. 99ECh. 5.7 - Let Z be a standard normal random variable and let...Ch. 5.7 - Prob. 101ECh. 5.8 - A firm purchases two types of industrial...Ch. 5.8 - Prob. 103ECh. 5.8 - Prob. 104ECh. 5.8 - Prob. 105ECh. 5.8 - In Exercise 5.9, we determined that...Ch. 5.8 - In Exercise 5.12, we were given the following...Ch. 5.8 - If Y1 is the total time between a customers...Ch. 5.8 - In Exercise 5.16, Y1 and Y2 denoted the...Ch. 5.8 - Suppose that Y1 and Y2 have correlation...Ch. 5.8 - Prob. 111ECh. 5.8 - In Exercise 5.18, Y1 and Y2 denoted the lengths of...Ch. 5.8 - A retail grocery merchant figures that her daily...Ch. 5.8 - For the daily output of an industrial operation,...Ch. 5.8 - Prob. 115ECh. 5.8 - Prob. 116ECh. 5.8 - A population of N alligators is to be sampled in...Ch. 5.8 - Prob. 118ECh. 5.9 - A learning experiment requires a rat to run a maze...Ch. 5.9 - Prob. 120ECh. 5.9 - Refer to Exercise 5.117. Suppose that the number N...Ch. 5.9 - The weights of a population of mice fed on a...Ch. 5.9 - Prob. 123ECh. 5.9 - The typical cost of damages caused by a fire in a...Ch. 5.9 - When commercial aircraft are inspected, wing...Ch. 5.9 - Prob. 126ECh. 5.9 - Prob. 127ECh. 5.10 - Let Y1 and Y2 have a bivariate normal...Ch. 5.10 - Prob. 129ECh. 5.10 - Prob. 130ECh. 5.10 - Prob. 131ECh. 5.10 - Prob. 132ECh. 5.11 - Prob. 133ECh. 5.11 - Prob. 134ECh. 5.11 - In Exercise 5.41, we considered a quality control...Ch. 5.11 - In Exercise 5.42, the number of defects per yard...Ch. 5.11 - In Exercise 5.38, we assumed that Y1, the weight...Ch. 5.11 - Assume that Y denotes the number of bacteria per...Ch. 5.11 - Prob. 139ECh. 5.11 - Prob. 140ECh. 5.11 - Let Y1 have an exponential distribution with mean ...Ch. 5.11 - Prob. 142ECh. 5.11 - Prob. 143ECh. 5 - Prove Theorem 5.9 when Y1 and Y2 are independent...Ch. 5 - Prob. 145SECh. 5 - Prob. 146SECh. 5 - Two friends are to meet at the library. Each...Ch. 5 - Prob. 148SECh. 5 - Prob. 149SECh. 5 - Prob. 150SECh. 5 - The lengths of life Y for a type of fuse has an...Ch. 5 - In the production of a certain type of copper, two...Ch. 5 - Suppose that the number of eggs laid by a certain...Ch. 5 - In a clinical study of a new drug formulated to...Ch. 5 - Prob. 155SECh. 5 - Refer to Exercise 5.86. Suppose that Z is a...Ch. 5 - Prob. 157SECh. 5 - Prob. 158SECh. 5 - Prob. 159SECh. 5 - Prob. 160SECh. 5 - Suppose that we are to observe two independent...Ch. 5 - Prob. 162SECh. 5 - Prob. 163SECh. 5 - Prob. 164SECh. 5 - Prob. 165SECh. 5 - Prob. 166SECh. 5 - Prob. 167SE
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.Similar questions
- prove the below Theorems (mean and variance of each) If Y is a chi-square random variable with v degrees of freedom, then u = E(Y) = v and o = V(Y) = 2v. Apply Theorem 4.8 with a = v/2 and B 2.arrow_forwardThe mean and variance of random variables X, Y, and Z are given as follows: ux = 1, ož = %3D In addition, we know that random variables X and Z are Hy = 6, oỷ = 1, and µz = 3, o} : NOT independent, and that ox+22-5 = 7. Based on the provided information, answer the following questions: %3D a) Find the covariance of X and Z (i.e., oxz). b) Calculate E(Y²). c) Find E(2X + Y – Z). d) Use the Chebyshev's theorem to give a lower bound for P(3 < Y < 9)? The proposed lower bound should be obviously larger than zero.arrow_forwardPreviously, you studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let x and y be random variables with means μx and μy, variances σ2x and σ2y, and population correlation coefficient ρ (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula. μw = aμx + bμyσ2w = a2σ2x + b2σ2y + 2abσxσyρ In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σxσyρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ2w reduces to the appropriate formula for independent variables. In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates.Do you have to be rich to invest in bonds and real estate? No, mutual fund…arrow_forward
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