Moment Generating Functions

Let Y be a random variable that takes on positive integers.

Where

a) Find the probability that

b) Find the moment generating function of .

c) Find the first and second moments

Attempt:

(A)

I don't really know what to do with it. I understand that it is an infinite geometric series that will sum to 1. But I'm not sure how to find Y = 0.

Using the formula for geometric series, I can find various p(k) in terms of p(0), but I don't know how to find p(0) itself.

(B)

I'm I on the right track? I'm not sure how to proceed after that.

(C)

That just involves finding the first and second derivative of the result I find in B, correct?

Re: Moment Generating Functions

You declare to be a variable that takes on positive integers. is not a positive integer. So, the probability that must be 0, right? But, that implies that the probability that must be 0 for all positive integers . Did you mean is a variable that can be any nonnegative integer?

Now, .

And in general,

Plugging that in, we have

.

Now, you know , so implies .

Does that help at all?

Re: Moment Generating Functions

Sorry, my mistake. It is indeed non-negative integers.

That definitively makes a lot more sense now.

Just the clarify my understanding, in the last step, the summation equals 2 because:

?

Then for part B, would this be the solution?

Thanks!

Re: Moment Generating Functions

Quote:

Originally Posted by

**RedXIII** Sorry, my mistake. It is indeed non-negative integers.

That definitively makes a lot more sense now.

Just the clarify my understanding, in the last step, the summation equals 2 because:

?

Then for part B, would this be the solution?

Thanks!

Yes for the reason why the summation equals 2. No for part B.

So,

Then