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1. 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?
2. (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

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

(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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