Given , 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?

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- February 27th 2009, 07:00 AMalthoughCDF of random summation of exponential radom variables
Given , 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?

- February 27th 2009, 07:52 AMmatheagle
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...

- February 27th 2009, 10:38 AMMoo
Hello,

It is possible to prove that , where 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 ^^