a. Explanation Calculation: From Exercise 5.87, it is obtained that V ( W 1 ) = 2 ν 1 + 2 ν 2 , V ( Y 1 ) = 2 ν 1 and E ( Y 1 ) = ν 1 . The random variable W 2 is expressed as W 2 = Y 2 + Y 3 . The expectation of the sum of the two given random variables can be written as, E ( Y 1 + Y 3 ) = E ( Y 1 ) + E ( Y 3 ) . The random variables of interest are Y 1 and Y 2 , each with a χ 2 -distribution, with respective degrees of freedom ν 1 and ν 2 . It is known that E ( Y a ) = 2 a Γ ( a + ν 2 ) Γ ( ν 2 ) , when Y has a χ 2 -distribution, with degrees of freedom ν and ν > − 2 a . Here, a = 1 . As a result, E ( Y 3 ) can be obtained as follows: E ( Y 3 ) = E ( Y 3 1 ) = 2 1 Γ ( 1 + ν 3 2 ) Γ ( ν 3 2 ) = 2 ( ν 3 2 ) Γ ( ν 3 2 ) Γ ( ν 3 2 ) = ν 3 . Similarly, E ( Y 3 ) = ν 3 . Hence, the value of E ( Y 1 + Y 3 ) is ν 1 + ν 3 [ = E ( Y 1 ) + E ( Y 3 ) ] . For the sum two given independent events, the variance can be written as, V ( Y 1 + Y 3 ) = V ( Y 1 ) + V ( Y 3 ) . The variance of Y 3 can be written as, V ( Y 3 ) = E ( Y 3 2 ) − [ E ( Y 3 ) ] 2 . The value of E ( Y 3 2 ) is calculated below ( a = 2 ) : E ( Y 3 2 ) = 2 2 Γ ( 2 + ν 3 2 ) Γ ( ν 3 2 ) = 4 ( 1 + ν 3 2 ) ( ν 3 2 ) Γ ( ν 3 2 ) Γ ( ν 3 2 ) = 2 ν 3 + ν 3 2 b. To determine Explain the reason that the answer of Part (a) is positive.

Mathematical Statistics with Applications

7th Edition

ISBN: 9780495110811

Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could co…

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a…

Videos

Question

Chapter 5, Problem 155SE

To determine

Calculate the value of Cov(W1,W2).

To determine

Explain the reason that the answer of Part (a) is positive.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5, Problem 154SE

Chapter 5, Problem 156SE

Students have asked these similar questions

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 = r1/2

Let 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 ?

A chi-squared random variable with ν > 0 degrees of freedom (χv2) has mgf M(t) = (1 − 2t) −ν/2 . Given that Z2 ∼ χ21, derive the mean and variance of Z2 using M(t). Confirm these results using the mgf of Z, namely MZ(t) = e1/2t2 .

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. 6E Ch. 5.2 - Let Y1 and Y2 have joint density function...Ch. 5.2 - Prob. 8E Ch. 5.2 - Prob. 9E Ch. 5.2 - An environmental engineer measures the amount (by...

Ch. 5.2 - Suppose that Y1 and Y2 are uniformly distributed...Ch. 5.2 - Prob. 12E Ch. 5.2 - Prob. 13E Ch. 5.2 - Prob. 14E Ch. 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. 18E Ch. 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. 24E Ch. 5.3 - Prob. 25E Ch. 5.3 - Prob. 26E Ch. 5.3 - Prob. 27E Ch. 5.3 - In Exercise 5.10, we proved that...Ch. 5.3 - Prob. 29E Ch. 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. 32E Ch. 5.3 - Suppose that Y1 is the total time between a...Ch. 5.3 - Prob. 34E Ch. 5.3 - Prob. 35E Ch. 5.3 - Prob. 36E Ch. 5.3 - Prob. 37E Ch. 5.3 - Let Y1 denote the weight (in tons) of a bulk item...Ch. 5.3 - Prob. 39E Ch. 5.3 - Prob. 40E Ch. 5.3 - Prob. 41E Ch. 5.3 - Prob. 42E Ch. 5.4 - Let Y1 and Y2 have joint density function f(y1,...Ch. 5.4 - Prob. 44E Ch. 5.4 - Prob. 45E Ch. 5.4 - Prob. 46E Ch. 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. 50E Ch. 5.4 - Prob. 51E Ch. 5.4 - Prob. 52E Ch. 5.4 - Prob. 53E Ch. 5.4 - Prob. 54E Ch. 5.4 - Prob. 55E Ch. 5.4 - In Exercise 5.12, we were given the following...Ch. 5.4 - Prob. 57E Ch. 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. 60E Ch. 5.4 - Prob. 61E Ch. 5.4 - Prob. 62E Ch. 5.4 - Let Y1 and Y2 be independent exponentially...Ch. 5.4 - Prob. 64E Ch. 5.4 - Prob. 65E Ch. 5.4 - Let F1(y1) and F2(y2) be two distribution...Ch. 5.4 - Prob. 67E Ch. 5.4 - Prob. 68E Ch. 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. 71E Ch. 5.6 - In Exercise 5.1, we determined that the joint...Ch. 5.6 - Prob. 73E Ch. 5.6 - Refer to Exercises 5.6, 5.24, and 5.50. Suppose...Ch. 5.6 - Prob. 75E Ch. 5.6 - Prob. 76E Ch. 5.6 - Prob. 77E Ch. 5.6 - Prob. 78E Ch. 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. 83E Ch. 5.6 - In Exercise 5.62, we considered two individuals...Ch. 5.6 - Prob. 85E Ch. 5.6 - Prob. 86E Ch. 5.6 - Prob. 87E Ch. 5.6 - Prob. 88E Ch. 5.7 - In Exercise 5.1, we determined that the joint...Ch. 5.7 - Prob. 90E Ch. 5.7 - In Exercise 5.8, we derived the fact that...Ch. 5.7 - Prob. 92E Ch. 5.7 - Suppose that, as in Exercises 5.11 and 5.79, Y1...Ch. 5.7 - Prob. 94E Ch. 5.7 - Prob. 95E Ch. 5.7 - Prob. 96E Ch. 5.7 - The random variables Y1 and Y2 are such that E(Y1)...Ch. 5.7 - Prob. 98E Ch. 5.7 - Prob. 99E Ch. 5.7 - Let Z be a standard normal random variable and let...Ch. 5.7 - Prob. 101E Ch. 5.8 - A firm purchases two types of industrial...Ch. 5.8 - Prob. 103E Ch. 5.8 - Prob. 104E Ch. 5.8 - Prob. 105E Ch. 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. 111E Ch. 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. 115E Ch. 5.8 - Prob. 116E Ch. 5.8 - A population of N alligators is to be sampled in...Ch. 5.8 - Prob. 118E Ch. 5.9 - A learning experiment requires a rat to run a maze...Ch. 5.9 - Prob. 120E Ch. 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. 123E Ch. 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. 126E Ch. 5.9 - Prob. 127E Ch. 5.10 - Let Y1 and Y2 have a bivariate normal...Ch. 5.10 - Prob. 129E Ch. 5.10 - Prob. 130E Ch. 5.10 - Prob. 131E Ch. 5.10 - Prob. 132E Ch. 5.11 - Prob. 133E Ch. 5.11 - Prob. 134E Ch. 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. 139E Ch. 5.11 - Prob. 140E Ch. 5.11 - Let Y1 have an exponential distribution with mean ...Ch. 5.11 - Prob. 142E Ch. 5.11 - Prob. 143E Ch. 5 - Prove Theorem 5.9 when Y1 and Y2 are independent...Ch. 5 - Prob. 145SE Ch. 5 - Prob. 146SE Ch. 5 - Two friends are to meet at the library. Each...Ch. 5 - Prob. 148SE Ch. 5 - Prob. 149SE Ch. 5 - Prob. 150SE Ch. 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. 155SE Ch. 5 - Refer to Exercise 5.86. Suppose that Z is a...Ch. 5 - Prob. 157SE Ch. 5 - Prob. 158SE Ch. 5 - Prob. 159SE Ch. 5 - Prob. 160SE Ch. 5 - Suppose that we are to observe two independent...Ch. 5 - Prob. 162SE Ch. 5 - Prob. 163SE Ch. 5 - Prob. 164SE Ch. 5 - Prob. 165SE Ch. 5 - Prob. 166SE Ch. 5 - Prob. 167SE

Knowledge Booster

$Statistics$

Learn more about

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.

Calculate the value of Cov ( W 1 , W 2 ) .

Solution Summary: The author explains how to calculate the value of Cov(W_1,