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**1. Homework Statement**

The pmf of a random variable X is given by f(x) = π(1 − π)

^{x}for x = 0, 1, ..., ∞, and 0 ≤ π ≤ 1.

a) Show that this function actually is a pmf.

b) Find E(X).

c) Find the moment generating function of X, MX(t) = E(e

^{tX}).

**2. The attempt at a solution**

My solution was done numerically in MATLAB, but I suppose that there is probably an analytical solution as well. My biggest issue is the interpretation of π.

[PLAIN]http://img401.imageshack.us/img401/2015/proofox.png [Broken]

For the second part, I also did this numerically, by solving the series:

[tex]\sum_{x}x \cdot p(x) = \sum_{x}x \cdot f(X=x)[/tex]

Which evaluates, also through MATLAB, to be: E(X) = 1, but I suspect there is probably an analytical method for this as well?

The part I am struggling the most with is the last bit. I can get my MGF down to:

[tex]\sum_{\forall x} e^{tx} \pi (1-\pi)^{x}[/tex]

But I am not sure how to get rid of the infinite summation. I tried an infinite geometric series, but it only holds true for:

[tex]|e^{t}(1-\pi)| < 1[/tex]

Which means that E(X) cannot be found with the MGF.

Any ideas?

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