Expected number of frog jumps There is frog jumping forward on a line. Each jump distance is random with a known cumulative distribution function $F$. What is the expected number of jumps to reach (or go beyond) distance $d$ from the origin?
If there is a Wikipedia or other web page describing the formula in terms of $F$, just give a (deep) link. I'd prefer to have the proof as well.

Let $X_i$ denote the distance of jump $i$ of the frog. Each $X_i$ is independent of one-another. The total distance at jump $n$ is then $D_n=X_1+\cdots+X_n$. Define $\tau$ for $D_\tau$, where $\tau$ is the first time such that $D_\tau\geq d$. You are interested in $E(\tau)$. The distribution of $D_n$ is the $n$-fold convolution of $F$'s: $F_n = (F\star)^n:=F\star F\star\cdots \star F$.
$P(\tau> n) = P(D_n<d)$
Now use $E(\tau)=\sum_{n=0}^\infty P(\tau> n)=\sum_{n=0}^\infty P(D_n<d)$
Which we can simplify when $F$ is continuous by noting $P(D_n<d)=P(D_n\leq d)=F_n(d)$. I'm not sure if there's any more simplification you can do without an explicit form for $F$. 
