Solution Found!
Refer to Exercise 3.156.a If W = 3Y, use the
Chapter 3, Problem 157E(choose chapter or problem)
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