Let F be a continuous c.d.f. satisfying F(0) = 0, and suppose that the distribution with c.d.f. F has the memory less property (5.7.18). Define (x) = log[1− F(x)] for x > .
a. Show that for all t, h > 0,
1− F(h) = 1− F(t + h)/1− F(t).
b. Prove that (t + h) = (t) + (h) for all t, h > 0.
c. Prove that for all t > 0 and all positive integers k and m, (kt/m) = (k/m)(t).
d. Prove that for all t, c > 0, (ct) = c(t).
e. Prove that g(t) = (t)/t is constant for t > 0.
f. Prove that F must be the c.d.f. of an exponential distribution.