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Probabilty help please!

If x1~geometric(0.2) and x2~(0.4) what is the moment generating function of 2x1+4x2 and does 2x1+4x2 have geometric distribution?
Original post by Harriettttttt1
If x1~geometric(0.2) and x2~(0.4) what is the moment generating function of 2x1+4x2 and does 2x1+4x2 have geometric distribution?


So the definition of the moment generating function is

Unparseable latex formula:

\displaymode M_X(t) = \mathbb{E}(e^{tX})



If you do the sums you will find that for a geometric distribution with parameter p, this is

Unparseable latex formula:

\displaymode M_X(t) = \frac{p e^t}{1 - (1-p)e^t}



(provided that you're using the geometric distribution with support on the strictly positive integers)

To answer the next part, how do you work out the moment generating function of aX + bY (a, b integer constants) from those of X & Y? You should know a theorem that helps you with this. Then finally, is the MGF that you obtain in the form of the MGF of some geometric distribution?

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