Let X1X2 be independent random variables each uniformly dist

Let X1,X2,?. be independent random variables each uniformly distributed on (0,1). Let N=min{n > = 1: sum i=1 to n Xi > 1}. (a) Find the probability mass function of N. (b) Find the expected value of N. (c) Find the variance of N. It is mostly part (a) that I need help with, but help on all three parts would be nice.

Solution

N = min sums

Hence X is the sum of x1+x2+...+xk where the sum is greater than 1 for the least k

a) Pmf of N = pmf of x1+pmf of x2+...+pmf of xk where the sum is greater than 1 for the least k

PMF of each Xi = 1,0<x<1

Hence pmf of N = k, 0<N<k where  the sum is greater than 1 for the least k

b) E(N) = E(x1)+...E(xk)

= k/2

c) Var of N = var(x1+x2+...+xk)

= k(b-a)²/12