k=0,...,n.

Where n is a positive integer, 0 < p < 1 , q = 1 - p.

How would I show the mgf of X is

Also, if

how would I derive the mgf of Y?

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- August 19th 2009, 05:01 AM #1

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- August 19th 2009, 05:17 AM #2
Hello,

The mgf of a discrete distribution is :

So here, we have :

Now just recall Newton's binomial formula :

As for the second one, if the Xi's are independent, then the mgf of Y is the product of the mgf of the Xi.

If they're identically distributed, this will just be

- August 19th 2009, 05:29 AM #3

- August 19th 2009, 02:49 PM #4

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- August 19th 2009, 04:24 PM #5

The point of using MGF's is that you hope to recognize the resulting one.

Here you added m INDEPENDENT binomials.

This techniques proves that the result is a binomial with same probability of success (p) and now the number of trials is nm.

The MGF of X+Y where they are indep bin(n,p) and bin(m,p), SAME p, different sample sizes is

proving that X+Y is bin(n+m,p)

So, for you the mean is (nm)p and the variance is (nm)pq.

- August 19th 2009, 05:33 PM #6

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- August 19th 2009, 05:44 PM #7

- August 19th 2009, 05:56 PM #8

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- August 20th 2009, 05:45 AM #9

- August 20th 2009, 06:01 AM #10