Q4 Let X Be A Continuous Non Negative Random Variable With Distribution Function F X And Probability Density Function F 1 (45.83 KiB) Viewed 98 times
Q4 Let X be a continuous non-negative random variable with distribution function F(x) and probability density function f(t) which is such that f(x) = f'(o) for all or € [0,00). Define the failure rate function (also called hazard function) as f(x) r() 1- F(x)' and the survival function as G(x) = 1 - F(). (a) Show that for a non negative continuous random variable X with failure rate function r, X has survival function G() = = exp (- [ r(uydu).
(b) Consider a certain type of organ in the human body that degrades gradually at a slow but steady rate. To model this, we suppose that its failure rate function is linearly increasing r(x) = ax for some a > 0. It has been observed that the median lifetime of the organ is 60 years. What is the probability that such an organ lasts for more than 100 years?
(c) Find the probability density functions of non negative continuous random vari- ables with failure rate functions (i) r(t) = 23, (ii) r(x) = 1/(1+1).
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