The lifetime of a certain component, in years, has the probability density function (pdf) fx ( x ) = 10 x20 x50 Two such components, whose lifetimes are independent, are available. As soon as the first component fails, it is replaced with the second component. Let X be a random variable that denotes the lifetime of the first component, and let Y denote the lifetime of the second component. (a) Find the joint pdf of the random variables X and Y. (b) Find the pdf of the sum, Z = (X + Y). (c) Find P(X $ 1|Y > 1). (d) Find P (X + Y S 2)