Can you help me with this probability question?

Posted on 2017-02-10 06:33:00 in Math by JAMES

Let X1, X2, . . . be iid. random variables each with density xe^(−x) for x > 0 and 0 otherwise. Let S0 = 0 and Sn : = X1 + · · · + Xn, and N(t) : = max{n : Sn < t}. 
(a) Determine the density of the random variable S2. I got ((x^3)*e^(-x))/6 for x>0 and 0 otherwise for this by doing a convolution which I think is right. 
(b) Find the mass function of the random variable N(t). To find the density Sn for n>2 we need to do a long computation so I assume that this is not the right way to go. I feel that using Poisson processes may be the way to go here but I'm not quite sure how. 
In this question I am told that if I have a good understanding, I should be able to do this problem without any computation but I don't get how. Maybe because the density of the Xi's is close to an exponential density?

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