1. [30 points] You are given the density function of a random variable X as: f(x)=1x for 05x51 _1 for 15x52 and 0 elsewhere. Throughout this question, suppose your computer only has the capacity to generate uniform random variables on [0, l]. (a) Derive the inverse-transform method to generate X. (b) Derive the acceptance-rej ection method to generate X , using a proposal given by the uniform distribution on [0, 2]