Hey guys I've got several questions about statistics. Here's the first one. 1. Suppose X is a discrete random variable that takes on the three values x1, x2, x3 with probabilities p1, p2, p3 respectively. Describe how you could generate a random sample from X if all you had access to were a list of numbers generated at random from the interval [0, 1]. 2. Suppose X is a continuous random variable with c.d.f. FX(x) = P(X x). To make things easier, suppose further that X takes on values in an interval [a, b] (do allow for a to be −1 and b to be +1). Let Y be the random variable defined by Y = FX(X). This looks strange, but is perfectly valid since FX is just a function, and you are allowed to take functions of random variables. Your problem: show that Y distributed U[0, 1]. Method: calculate the c.d.f. Y , FY (y) = P(Y y), for all real y. Replace Y with FX(X), and consider when you can take the inverse of FX. Taking the inverse of FX is not always possible, in particular, for x < a and x > b. Consider those cases separately. You will find that the c.d.f. of Y is 0 for y < 0, y for 0 y 1, and 1 for y > 1, so indeed Y distributed U[0, 1]. An important application of this fact is that if u is a random selection from the interval [0, 1], F−1 X (u) is a random selection from X.