# Distribution of the summation of k random variables and k is also variable

We have a set of positive random variables $\boldsymbol X=\{X_1,X_2,\ldots\}$, where $X_1,X_2,\ldots,$ are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for Xi are known in advance.

$K$ is an integer random variable with CDF $F_K(k)$ and PDF $f_K(k)$.

Define $S_n=\sum_{i=1}^n X_i$, then what is the distribution of $S_K$?

This would be easy if $K$ is constant (but it isnt't). This means that we don't know (yet) how to compute $P(S)$, but we know how to compute $P(S | K)$ ("if $K$ were a constant"). Well, then use conditional probabilities:
$$P(S) = \sum_K P(S,K) = \sum_K P(S|K) P(K)$$
• Is there any simple way to compute $P(S\mid K)$? Because this involves $K$-fold convolution of $F(x)$ or $f(x)$. This is what I try to avoid. – Bloodmoon Jul 4 '15 at 11:24