Question

As a general comment, remember that showing two random variables have the same CDFor PDF is sufficient for showing that they have the same distribution. a) First, let us see how to generate an exponential random variable with a uniform random vari- (b) Let M.N2 ~ y (0, î ), where Ni and N2 are independent. Prove that NỈ + N: ~ Expo( 1 /2). able. Let U1~Uniform(0, 1). Prove that-InU1Expo(1). Hint: You may use the fact that over a region R, if we convert to polar coordinates (x,y) -> (r, 8), then the double integral over the region R will be f(x,y) dxdy- 11 f(rcos θ , r sin0). rdrd8. (c) Suppose we have two uniform random variables, U1 and U2. How would you transform these two random variables into a normal random variable with mean 0 and variance 1? Hint: What part (b) tells us is that the point (Ni,N2) will have a distance from the origin that is distributed as the square root of an exponential distribution. Try to use U1 to sample the radius, and then use U2 to sample the angle.

Answer 1