Hi,

In a derivation for the expectation of an RV with a geometric distribution I have the following:

$\displaystyle E(X) = \sum^{\infty}_{k=r} k \binom{k-1}{r-1}p^{r} (1-p)^{k-r} = \frac{r}{p} \sum^{\infty}_{k=r} \binom{k}{r} p^{r+1} (1-p)^{k-r} = \frac{r}{p} $

I have several questions.

1. What sorcery took place with the binomial coefficient and k, such that k disappeared, r appeared as a factor and its now $\displaystyle \binom{k}{r}$ rather than$\displaystyle \binom{k-1}{r-1}$ ?

2. Presumably the logic is that $\displaystyle \sum^{\infty}_{k=r} \binom{k}{r} p^{r+1} (1-p)^{k-r} = 1$. Can anyone show me how to prove this and what class of distribution is this, not binomial right?

Many thanks in advance. MD.