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tangodirt
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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(etX).
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
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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