# Summation of geometric number of iid exponentially distributed random variables

Hello, I am having difficulty approaching this problem:

Assume that K, Z_1, Z_2, ... are independent.
Let K be geometrically distributed with parameter success = p, failure = q.
P(K = k) = q^(k-1) * p , k >= 1

Let Z_1, Z_2, ... be iid exponentially distributed random variables with parameter (lambda).
f(z) =
(lambda)*exp(-(lambda)x) , x >= 0
0, otherwise

Find the cdf of Z_1 + Z_2 + ... + Z_K

I think there is some relation to the Gamma function here, but I'm not quite sure how...

Any hints/suggestions?

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Well, I would try something like this. Let S be your "random number of random variables", ie. Z1+Z2+...+ZK

P(S<X) = sum_n P(S<X | K=n) * P(K=n)

Then analyze P(S<X | K=n) by finding the PDF or CDF for a random variable that is the sum of n exponential random variables. You could use the result that the resulting distribution function is the convolution of the n distribution functions.

After you have found P(S<X | K=n), you can perform the sum of n.

Torquil