5 Short Answer

(a) Let X be uniform on the interval [0, 2], and define Y = 2X +1. Find the PDF, CDF, expectation, and variance of Y.

(b) Let X and Y have joint distribution

f(x, y) = { cxy + 1/4 x [1, 2] and y [0, 2]

{0 else

Find the constant c. Are X and Y independent?

(c) Let X Exp(3). What is the probability that X [0, 1]? If I define a new random variable Y = X, for each k N, what is the probability that Y = k? Do you recognize this (discrete) distribution?

(d) Let X_i Exp(_i)for i = 1, ..., n be mutually independent. It is a (very nice) fact that min(X₁, ..., X_n) Exp(). Find .