Suppose that Y is a random variable with moment-generating

Chapter 3, Problem 156E

(choose chapter or problem)

Get Unlimited Answers
QUESTION:

Suppose that \(Y\) is a random variable with moment-generating function \(m(t)\).

a What is \(m(0)\)?

b If \(W=3Y\), show that the moment-generating function of \(W\) is \(m(3t)\).

c If \(X=Y?2\), show that the moment-generating function of \(X\) is \(e^{-2 t} m(t)\).

Questions & Answers


(1 Reviews)

QUESTION:

Suppose that \(Y\) is a random variable with moment-generating function \(m(t)\).

a What is \(m(0)\)?

b If \(W=3Y\), show that the moment-generating function of \(W\) is \(m(3t)\).

c If \(X=Y?2\), show that the moment-generating function of \(X\) is \(e^{-2 t} m(t)\).

ANSWER:

Step 1 of 3:

(a)   Assume that \(Y\) is a random variable with moment-generating function \(m(t)\)

We need to find the value of \(m(0)\)

The moment-generating function \(m_{Y}(t)\) for a random variable \(Y\) is defined to be,

\(m_{Y}(t)=E\left(e^{t Y}\right)\)

(1)

We are asked to find \(m(0)\), hence put \(t=0\), into the equation (1),

\(\begin{array}{l} m_{Y}(0)=E\left(e^{0 \times Y}\right) \\ m_{Y}(0)=E\left(e^{0}\right)=E(1)=1 \end{array}\)

[since \(E(C)=C, \text { where } C \text { is a constant }\)]

Hence the value of \(m(0)\) is \(1\)

Add to cart

Reviews

Review this written solution for 31605) viewed: 334 isbn: 9780495110811 | Mathematical Statistics With Applications - 7 Edition - Chapter 3 - Problem 156e

Thank you for your recent purchase on StudySoup. We invite you to provide a review below, and help us create a better product.

Textbook: Mathematical Statistics with Applications

Click to rate

Write a review below (optional):

Submit Review
×

Thanks for your review!

Think of all the students you've helped. Nice job!


Study Tools You Might Need

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!

Forgot password?
Register Now

×

Register

Sign up for access to all content on our site!

Or login if you already have an account

×

Reset password

If you have an active account we’ll send you an e-mail for password recovery

Or login if you have your password back