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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 $S_K$?

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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:

$$P(S) = \sum_K P(S,K) = \sum_K P(S|K) P(K)$$

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  • $\begingroup$ 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. $\endgroup$ – Bloodmoon Jul 4 '15 at 11:24
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    $\begingroup$ No, I don't think you can avoid the convolution. Recall only that if you use CF (characteristic functions) it might be simpler, because convolutions translate to products. $\endgroup$ – leonbloy Jul 4 '15 at 11:36
  • $\begingroup$ I see, mathematically CF provides a way to avoid direct convolution calculation, but convert the CDF or PDF to Fourier Transform so that convolution can be translated to products, which is simpler. Then we can convert back by inverse Fourier Transform. Is this correct? (Of course CF also provides other functionalities.) $\endgroup$ – Bloodmoon Jul 4 '15 at 11:56

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