Solution Found!

Statistics / Mathematical Statistics with Applications 7 / Chapter 4 / Problem 137E

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

Change Question

Show that the result given in Exercise 3.158 also holds

Chapter 4, Problem 137E

(choose chapter or problem)

Get Unlimited Answers

QUESTION:

Problem 137E

Show that the result given in Exercise 3.158 also holds for continuous random variables. That is, show that, if Y is a random variable with moment-generating function m(t) and U is given by U = aY + b, the moment-generating function of U is etbm(at). If Y has mean μ and variance σ 2, use the moment-generating function of U to derive the mean and variance of U .

Reference

If Y is a random variable with moment-generating function m(t) and if W is given by W = aY + b, show that the moment-generating function of W is etbm(at).

Questions & Answers

QUESTION:

Problem 137E

Show that the result given in Exercise 3.158 also holds for continuous random variables. That is, show that, if Y is a random variable with moment-generating function m(t) and U is given by U = aY + b, the moment-generating function of U is etbm(at). If Y has mean μ and variance σ 2, use the moment-generating function of U to derive the mean and variance of U .

Reference

If Y is a random variable with moment-generating function m(t) and if W is given by W = aY + b, show that the moment-generating function of W is etbm(at).

ANSWER:

Solution 137E

Step1 of 2:

Let us consider a random variable Y it has mean $\mu{}$ and variance ${\sigma{}}^{2}$ . If Y is a random variable with moment-generating function m(t).

Also we have U = aY + b.

We need to find the moment generating function of U and also we need to derive mean and variance of U.

Step2 of 3:

Let,

${m}_{u}^{\ }(t)=E({e}^{tU})$

Substitute U value in above equation we get,

$=E({e}^{t(aY+b)})$

$=E({e}^{bt}{e}^{aYt})$

$={e}^{bt}m(at)$

Therefore moment generating function of U is ${m}_{u}^{\ }(t)={e}^{bt}m(at)$ .

Study Tools You Might Need

Chapter: 4 Problem: 3Q

Introductory Chemistry 5
Describe Rutherfords gold foil experiment and the results of that experiment. How did these results refute the plum pudding model of the atom?

Chapter: 7 Problem: 73P

Introductory Chemistry 5
Write balanced complete ionic and net ionic equations for each reaction. (a) \(\mathrm{AgNO}_{3}(a q)+\mathrm{KCl}(a q) \rightarrow \mathrm{AgCl}(s)+\mathrm{KNO}_{3}(a q)\) (b) \(\mathrm{CaS}(a q)+\mathrm{CuCl}_{2}(a q) \rightarrow \mathrm{CuS}(s)+\mathrm{CaCl}_{2}(a q)\) (c) \(\mathrm{Na

Chapter: 2 Problem: 3P

Principles of General Chemistry 2
Problem 3P Define Physical Property and chemical property. Identify each type of property in the following statements: (a) Yellow-green chlorine gas attack silvery sodium metal to form white crystal of sodium chloride(table salt). (b) A magnet separates a mixture of black iron shavings and white sand

Chapter: 6 Problem: 96P

Principles of General Chemistry 2
Problem 96P Aqueous sulfurous acid (H2SO3) was made by dissolving 0.200 L of sulfur dioxide gas at 19C and 745 mmHg in water to yield 500.0 mL of solution. The acid solution required 10.0 mL of sodium hydroxide solution to reach the titration end point. What was the molarity of the sodium hydroxide s

Chapter: Problem: 34

Fundamentals of Physics Extended 9
Two types of barrel units were in use in the 1920s in the United States. The apple barrel had a legally set volume of 7056 cubic inches; the cranberry barrel, 5826 cubic inches. If a merchant sells 20 cranberry barrels of goods to a customer who thinks he is receiving apple barrels, what is the discr

Chapter: 2 Problem: 8

Principles of Economics 6
What are the two subfields into which economics is divided? Explain what each subfield studies.

Chapter: 2 Problem: 14

Biology
Describe how a forest ecosystem might be different without the presence of decomposers and detritivores.

Chapter: 7 Problem: 15

Foundations in Microbiology 10
Explain what it means to be an obligate intracellular parasite. Name three groups of obligate intracellular parasites.

Not The Solution You Need? Search for Your Answer Here:

×

Login

Login or Sign up for access to all of our study tools and educational content!

×

Register

Sign up for access to all content on our site!

Or login if you already have an account