Refer to Exercise 3.156.a If W = 3Y, use the

Chapter 3, Problem 157E

(choose chapter or problem)

Get Unlimited Answers
QUESTION:

Refer to Exercise 3.156.

a If \(W=3 Y\), use the moment-generating function of \(W\) to show that \(E(W)=3 E(Y)\) and \(V(W)=9 V(Y)\).

b If \(X=Y-2\), use the moment-generating function of \(X\) to show that \(E(X)=E(Y)-2\) and \(V(X)=V(Y)\).

Questions & Answers

QUESTION:

Refer to Exercise 3.156.

a If \(W=3 Y\), use the moment-generating function of \(W\) to show that \(E(W)=3 E(Y)\) and \(V(W)=9 V(Y)\).

b If \(X=Y-2\), use the moment-generating function of \(X\) to show that \(E(X)=E(Y)-2\) and \(V(X)=V(Y)\).

ANSWER:

Step 1 of 2

(a)

By referring to exercise 3.156.

If  use the moment-generating function of  to show that  is  and  is  

We need to prove the mean of random variable   is

First, let find

We know the moment-generating function  for a random variable

……….(1)

Similarly, we can write the moment-generating function  for a random variable

………..(2)

We have given  substitute into the equation (2),

 ……..(3)

By comparing equation (1) and equation (3), we can write,

…….(4)

Hence the moment-generating function of  is

If  exists, then for any positive integer ,

…….(5)

In other words, if you find the  derivative of  with respect to  and then set t = 0, the result will be

 or  is the  derivative of  Because

It follows that

(6)

(7)

Setting  into equation (6) and (7)

………(8)

……….(9)

We know the following relation of variance with mean,

……….(10)

Hence using equation (5), we can find

 

[from equation (8) ]

Hence the mean of random variable   is

We also need to prove the variance of random variable   is

[from equation (9) ]

[]

Hence the variance of random variable   is

Add to cart


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