**Problem 5a**

Suppose that and are independent random variables with the following moment generating functions:

Find the probability .

**Problem 5b**

Suppose that and are independent random variables with the following moment generating functions:

Find the probability .

* Solution is found below*.

**Solution to Problem 5a**

The mgf is that of a binomial distribution with and . The mgf is that of a Poisson distribution with parameter .

**Answer to Problem 5b**

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Part B — Help! I cannot figure out how to get the probability distribution for X. A hint would be very appreciated!

The skill to practice here is to recognize moment generating functions of some familiar distributions. For 5-B, one mgf is that of a geometric distribution. Hope this helps.

I still can’t see it. I see that Y is a discrete distribution once one expands the square. But X…? I assume this is the one you are saying is a geometric. My form for geometric is p/(1-qe^t) (This is using the E[x] = p/q form of the geometric.) The 3 in the denominator is what is throwing me off… Am I missing some algebra step that gets it to the desired form?

The mgf can be transformed as:

.

Ok, I got it now… I didn’t see the algebra on the geometric distribution (as I had guessed was my problem). Thanks for the hint… and the great blog. 🙂