*48. Let £ / , , /> 1, denote independent uniform (0, 1) random variables.

For 0 < a < 1, define TV by

Í = mm{n :Ul + · · · + Un> a]

(a) Show by induction that an P[N> n] = —

n\

(b) Prove that E[N] = ea, and conclude that the expected number of random numbers needed until their sum exceeds 1 is equal to e.