大學題目~renewal process(10mark!)

1. Suppose that the interarrival times in a renewal process [Xt, t >= 0} are uniform over (0 1).
(a) Obtain the renewal function m(t) for 0 <= t <= 1.
(b) Obtain m(t) for 1 < t <= 2.

2. (a) Obtain the survival and probability density functions for a distribution whose hazard function is 入(s) = ns^(n-1), s > 0 and n >= 1.
(b) Suppose X1 and X2 are two independent and identically distributed random variables with hazard function 入(s), s > 0. Obtain, in terms of 入(s) the hazard function of min(X1,X2). Reconcile your results with your knowledge of the exponential distribution.