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4. (20 points) Consider a continuous random variable X with cdf F(x) and pdf f(x). The hazrd rate is defined by: r(a) = lim P(x 0. (a) Show that r(x) = f(x) / (1 - F(x)). (b) If r(x) = c, where c is a constant, show that the underlying distribution is exponential. Hence, exponentail distributions have constant failure rate over all time. (c) If r(x) = ca, where c and b are positive constants, show that X has a Weibull distri- bution, i.e., Cab+1 f (a) = cx exp for 0