Determine the hazard rate function for the random variable

Exercise 8 (#1.22). Let v be a measure on a o-field F on ? and f and g be Borel functions with respect to F. Show that (i) if f fdy exists and a e R, then [(af)dy exists and is equal to a ] fdy; (ii) if both / fdu and / gdy exist and f fdu + f gdy is well defined, then (f + g)dv exists and is equal to ] fdy + ] gdv. Note. For integrals in calculus, properties such as J(af)dy =a f fdy and [(f + g)dv / fdy + / gav are obvious. However, the proof of them are complicated for integrals defined on general measure spaces. As shown in this exercise, the proof often has to be broken into several steps: simple functions, nonnegative functions, and then general functions.5 Random variables X and yare independent random variables. The random variable x has chi- square distribution with 1 degree of freedom . The sum of random variables Xty has a chi- square distribution with 3 degrees of freedom. a,) The random variable yhas a distribution. I, normal II. Chebychev's V. F IL. binomial II. chi-square II. Not enough inform. b. The random variable yhas degrees of freedom I. D II. 2 V.4 I.1 IV.3 VI Not enough informationProb. 5 Let X be an exponentially distributed random variable with probability density function (PDF) given by Sie-2 fx (x) = 10, x 2 0 otherwise Consider the random variable Y = VX . (a) Determine the hazard rate function for the random variable Y. (b) Give an algorithm for generating the random variable Y from a standard uniform random variable U. (c) Choose a value for the parameter 1 so that the variance of the random variable Y is twice its mean.Prob. 5 (20 points) (a) Let X be a random variable with pdf given by fx (x) = ex, x > 0. Find the CDF of the random variable Y = e and, hence, give an algorithm to generate a random variable Y. (b) A random variable is generated from X = FX1(U) = [-In(1 - U)]1/8, where U is a standard uniform random variable. Find the PDF of the random variable X