## probability generating function

Hi All,

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

Gx(t)=(3t/4)/(1-t/4)
Gy(t)=((1+t)t^2)/2

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.