Let X and Y be independent integrable random variables on a probability space and f be a nonnegative convex function. Show that E[f(X +Y)] ≥ E[f(X +EY)]

A) 0.4625 B) 0.3375 C) 0.600 D) 0.4355

Let X be a random variable on a closed and bounded interval [a, b]. Let g(x) be a convex function. Prove that                                 g(E(X)) ≤ E (g(X)

Let X be a point randomly selected from the unit interval [0, 1]. Consider the random variable Y = (1-X)-¹/2 (a) Sketch Y as a function of X. (b) Find and plot the cdf of Y. (c) Derive the pdf of Y. (d) Compute the following probabilities: P(Y > 1), P(3 < Y < 6), P(Y ≤ 10).