We say that X has the ‘lack-of-memory property’ since, if we are given that X − m > 0, then the distribution of X − m is the same as the original distribution of X. Show that the geometric distribution is the only distribution concentrated on the positive integers with the lack-of-memory property.

6. The random variable N takes non-negative integer values. Show that E(N) =

∞X k=0 P(N > k)

provided that the series on the right-hand side converges.