Prove $p(x,y) = q^2p^{y-2}$ is a probability mass function Question: Prove $p(x,y) = q^2p^{y-2}$ is a pmf
where $q = 1 - p$ $$x = 1,2,3,...,y-1$$ $$y = 2,3,..., \infty $$
My attempt:
I am trying to prove $p(x,y)$ must add up to 1
$$\sum_{y=2}^{\infty} \sum_{x=1}^{y-1} q^2p^{y-2} = \sum_{y=2}^{\infty} q^2p^{y-2} \sum_{x=1}^{y-1} 1 = \sum_{y=2}^{\infty} q^2p^{y-2} (y-1) $$
to me this is kinda similar to a summation of a binomial however there is no way I can get $y-1$ to equal $y \choose 2$ (also, the range of summation is not the same as of binomial's!!!)
I am stuck there and any help would be appreciated. Thanks!

I don't think your question is well formed but it's almost certain that the problem can be done using the formula for the sum of a geometric series
$$
\sum_{y=Y}^\infty p^y=p^Y{1\over 1-p}
$$
if $p<1$.
