Let $X$ and $Y$ be independent geometric random variables with the same parameter $p$. What is the probability that $X+Y$ is odd?
Based on the provided solutions, I can follow through until this part which confuses me:
$$P(A) = \sum_{k = 2,\ k\; \text{even}}^\infty \;p(1-p)^{k-1} = p\sum_{i=1\; }^\infty (1-p)^{2i-1} $$
My question:
1) The lower summation bound states k=2 and k even. I do not understand why this is necessary ? if k is starting at 2 then k is automatically even, no?
See full solution below:
