Mathematical Statistics with Applications
Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer
Publisher: Cengage Learning
Question
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Chapter 5, Problem 163SE

a.

To determine

Prove that joint density function of Y1and Y2 is same as given in Exercise 5.65.

a.

Expert Solution
Check Mark

Explanation of Solution

Calculation:

In Exercise 5.65 the joint density is given as follows:

f(y1,y2)={[1α{(12ey1)(12ey1)}]ey1y2,y10,y200,otherwise,

Where marginal densities of Y1and Y2 follow exponential distribution with mean 1 and 1α1.

It is known that the cumulative density function of Y1 is F1(y1)=1ey1 and the cumulative density function of Y2 is F2(y2)=1ey2.

Now, substitute F1(y1)=1ey1 and F2(y2)=1ey2 in the given joint cumulative density function.

That is,

F(y1,y2)=(1ey1)(1ey2)[1α{1(1ey2)}{1(1ey2)}]=(1ey1)(1ey2)[1αey1ey2]

Hence, the joint probability density function is obtained as follows:

f(y1,y2)=2F(y1,y2)y1y2=2[(1ey1)(1ey2)[1αey1ey2]]y1y2=2[(1ey1)(1ey2)]y1y2α2[(1ey1)(1ey2)ey1ey2]y1y2=ey1ey2ey1ey2{α(12ey1)(12ey2)}=ey1y2ey1y2{α(12ey1)(12ey2)}=[1α{(12ey1)(12ey1)}]ey1y2

Hence, it is proved that the joint density function of Y1and Y2 is same as given in Exercise 5.65.

b.

To determine

Evaluate F(y1,y2) for any α where 1α1.

b.

Expert Solution
Check Mark

Answer to Problem 163SE

The joint cumulative density function of Y1and Y2 is F(y1,y2)=y1y2[1α(1y1)(1y2)],1α1.

Explanation of Solution

Calculation:

Consider that Y1and Y2 follow Uniform distribution over the interval (0,1).

Hence, the F1(y1)=y1and F2(y2)=y2.

Now, substitute F1(y1)=y1and F2(y2)=y2 in the given joint cumulative density function.

That is,

F(y1,y2)=(y1)(y2)[1α{1(y1)}{1(y2)}]=y1y2[1α(1y1)(1y2)]

Hence, the joint cumulative density function of Y1and Y2 is F(y1,y2)=y1y2[1α(1y1)(1y2)],1α1.

c.

To determine

Obtain the joint density function associated with the distribution function that is obtained in Part (b).

c.

Expert Solution
Check Mark

Answer to Problem 163SE

The joint density function is,

f(y1,y2)={1α(12y1)(12y2),0y11,0y210,Otherwise.

Explanation of Solution

From Part (b), the joint cumulative density function of Y1and Y2 is obtained as F(y1,y2)=y1y2[1α(1y1)(1y2)],1α1.

Hence, the joint probability density function is obtained as follows:

f(y1,y2)=2F(y1,y2)y1y2=2[y1y2[1α(1y1)(1y2)]]y1y2=2(y1y2)y1y2α2[y1y2(1y1)(1y2)]y1y2=1α2[(y1y12)(y2y22)]y1y2=1α(12y1)(12y2)

Thus, the joint density function is,

f(y1,y2)={1α(12y1)(12y2),0y11,0y210,Otherwise.

d.

To determine

Provide two specific and different joint densities that yield marginal densities for Y1and Y2 both of which are Uniform over the interval (0,1).

d.

Expert Solution
Check Mark

Explanation of Solution

Calculation:

From Part (c), the density function is obtained as follows:

f(y1,y2)={1α(12y1)(12y2),0y11,0y210,Otherwise,

Where marginal densities of Y1and Y2 follow Uniform over the interval (0,1).

The marginal density function does not depend on the values of α.

Thus, as α(0,1), one can take any two values of α within that range to obtain two different densities.

Consider α=0.5 and substitute α=0.5 in the given probability density function.

f(y1,y2)={1+0.5(12y1)(12y2),0y11,0y210,Otherwise

Consider α=0.5 and substitute α=0.5 in the given probability density function.

f(y1,y2)={10.5(12y1)(12y2),0y11,0y210,Otherwise

Hence, these two specific and different joint densities yield marginal densities for Y1and Y2 that are both Uniform over the interval (0,1).

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Chapter 5 Solutions

Mathematical Statistics with Applications

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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