1. Hazard rate or failure rate: Suppose X is a non-negative continuous random variable with probability distribution function (pdf) f(a) and cumulative distribution function (edf) F(x). The function f(t) 1 F(t) is called the hazard rate or failure rate function of F and can be thought of the conditional probability intensity that a t-unit old item will fail. That is, if X represents the lifetime of some item, and it has survived t time t) for t 20 units, then A(t) gives the probability that it will not survive an additional short time dt. (a) Show that the hazard or failure rate for an exponential distribution with parameter A, is a constant. That is A(t) Ca constant. What is C (b) Let X be a positive, continuous random variable with probability density function { if > 0 2 (1+) 0, f (x) = otherwise (c) Compute the failure rate of X and determine its limit as t oo. (d) Show that in general, (u)duIn (1 F(t), justifying each step, and then deduce that F(t) 1-eJo a(u)du (e) Use (d) to answer the question: The hazard or failure rate for lung cancer for at years old male smoker is such that t)0.027+0.00025(t 40)2, for t 40 Assume that a 40 years old smoker survives all other hazards, what is the probability that he survives to age 55, without contracting lung cancer?