Question

please show all work thank you a) Derive the expression for the moment generating function for...

please show all work thank you

a) Derive the expression for the moment generating function for the standard normal random variable Z.

b) Suppose Z has a standard normal distribution. Use the method of distribution functions or the method of transformations to find the density function of Y = σZ + µ and identify the distribution.

c). Using the moment generating function for Z obtained in problem a, find the moment generating function for Y = σZ + µ.

Math   Statistics-And-Probability   
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Homework Answers

Answer #1

a)

The pdf of Z is

The MGF is

Last integral is pdf of normal distribution with mean t and standard deviation 1. So

b)

Since so

Now

The pdf of y will be

It is pdf of normal distribution with mean and standard deviation .

c)

Now

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