Most sources say it's $\frac{r(1-p)}{p^2}$ but I can't seem to get it. What am I doing wrong?
Let $X$ =$X_1 + X_2 + \cdots + X_r$.
$Var(X) = E(X^2) - E(X)^2$
$Var(X_1 + X_2 + \cdots + X_r) = E((X_1 + X_2 + \cdots + X_r)^2) - (E(X_1 + X_2 + \cdots + X_r))^2$
Then,
$E((X_1 + X_2 + \cdots + X_r)^2)$
$ = E(\sum_{i,j} X_iX_j)$
$ = \sum_{i, j} E(X_iX_j)$
$ = \sum_{i, j} E(X_i)E(X_j)$
since $X_i$ and $X_j$ are independent for all $i, j$.
$ = (r^2) E(X_i)E(X_j)$
and since $E(X_i) = \frac{1}{p}$ for all $i$, then our expression equals $\frac{r^2}{p^2}$
which is clearly wrong.