# CDF of random summation of exponential radom variables

• Feb 27th 2009, 06:00 AM
although
CDF of random summation of exponential radom variables
Given $\displaystyle X=\sum_{i=1}^N Z_i$, where N is a random variable, and Z_i are i.i.d. exponential random variable with rate 1, N and Z_i are independent. What's the relationship between the CDF of X and the CDF of N? Do we have any closed form relations or bounds between them?
• Feb 27th 2009, 06:52 AM
matheagle
I don't think I've ever seen this b4, but you will need to know the relationship between the Z's and N, like if they are independent. Then I would think you'd attack the problem via...
$\displaystyle P\{Z_1+\cdots +Z_N\le a\}=P\{Z_1+\cdots +Z_n\le a|N=n\}P\{N=n\}$
• Feb 27th 2009, 09:38 AM
Moo
Hello,

It is possible to prove that $\displaystyle G_{X}(s)=G_N \circ G_{Z_1}(s)$, where $\displaystyle G_X$ is the probability generating function of the r.v. X.

I haven't found a relationship between the CDF and the PGF, but maybe you can ^^