Solution Found!
Example 4.16 derives the moment-generating function for Y
Chapter 4, Problem 138E(choose chapter or problem)
Example 4.16 derives the moment-generating function for \(Y-\mu\), where 𝑌 is normally distributed
with mean \(\mu\) and variance \(\sigma^{2}\).
a Use the results in Example 4.16 and Exercise 4.137 to find the moment-generating function
for 𝑌.
b Differentiate the moment-generating function found in part (a) to show that \(E(Y)=\mu\) and \(V(Y)=\sigma^{2}\).
Equation Transcription:
Text Transcription:
Y-mu
mu
2
E(Y)=mu
V(Y)=sigma^2
Questions & Answers
QUESTION:
Example 4.16 derives the moment-generating function for \(Y-\mu\), where 𝑌 is normally distributed
with mean \(\mu\) and variance \(\sigma^{2}\).
a Use the results in Example 4.16 and Exercise 4.137 to find the moment-generating function
for 𝑌.
b Differentiate the moment-generating function found in part (a) to show that \(E(Y)=\mu\) and \(V(Y)=\sigma^{2}\).
Equation Transcription:
Text Transcription:
Y-mu
mu
2
E(Y)=mu
V(Y)=sigma^2
ANSWER:
Solution 138E
Step1 of 3:
Let us consider a random variable Y it is normally distributed with mean and
Also we have U = Y - .
Here our goal is:
a). We need to find the moment-generating function for Y .
b). We need to show that E(Y ) = μ and V (Y ) = σ 2.
Step2 of 3:
a).
Let,
Then,
We know that the moment generating function of U is:
Use then the moment generating function of Y is given by: