2. Let U be a continuous random variable having a uniform distribution on the interval [0, 1]. Observed values having this distribution can be obtained from a computer's random number generator (a device that is able to select a number at random in the interval [0, 1] in such a way that each number in the interval is, for all practical purposes, equally likely to be selected). Let A be a positive constant, and let X be a new random variable defined as a function of U as follows: x= In(1 In(1 - U). Show that X has the exponential distribution with parameter A by following the steps below. (a) Find the probability density function of U and draw a sketch (this is just a uniform pdf). (b) Show that the cumulative distribution function of X is the one of the exponential distribution with parameter A by showing that if < 0 Fx(x) = P(X ≤2)= {1-220 To do this you have to rewrite the probability P(XS) in terms of the variable U (rewrite the inequality X in terms of the variable U), and then use the density function of U you found in (a) to calculate the probability. Notice that this proves that you can use a random number generator to simulate a random variable that has an exponential distribution with parameter A: you can ask the computer to produce a random number u and then calculate x -- In(1-u) to obtain an observed value of an exponential random variable with parameter A.