I'm new to the forum. I'm trying to learn a bit of statistics on my own (as a pastime) but I'm stuck with this problem:
I have two independent random variables X and Y with probability generating functions
and I have to find P(3X-2Y=12)
The problem I have is that I can't find an easy way to work out the coefficients I need in Gs(t), where S = 3X - 2Y.
I know that X has a geometric distribution and that the only values that 2Y can have are 4, 5 and 6.
If I'm not mistaken, the probability generating function for Gs(t) should be:
(3t/4)^3 * (1 + t/4 + (t/4)^2 + (t/4)^3 + ...)^3 * (4/t^4 + 2/t^5 + 4/t^6)
Hence, I have to find the coefficients for t^13, t^14 and t^15 in the cube of the geometric progression. But I can't think of any easy way of doing this. So I suppose there's either an alternative way of solving this problem or there's a formula I'm not aware of which gives me the coefficients.
I'd greatly appreciate any pointers! Thanks.
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