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...
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?