5. A continuous random variable X has the following cumulative distribution function (cdf). 0, Fx(x) = (x – 1)², 1 3 5.2 Give the probability density function (pdf) of X, fx(x). 5.3 Give the mean of X, E[X]. 5.4 Give the variance of X, Var[X].

3) Suppose X is a discrete variable that has the following pr function (pdf) f (X) 1. 0.40 0.20 3 0.15 4. 0.25 a) Calculate the cumulative distribution function (the b) Find the expected value, showing your work: E(X) c) Find the variance, showing your work: Var (X)

12. i. State Central limit theorem and tell what is the main theme of it. ii. Show that if the random variables (X, Y) follow the probability density of a bivariate normal distribution and if the correlation p = 0, X and Y are independent. (Show that f(x, y) = g(x) h(y))

5.3-2 The random variable X has probability density function - {2(1-x). 0, 5.3-3 f(x) = 0 b probability histogram of the data. 2(1-x). 0≤x≤ 1, otherwise. Graph the probability density function of X. Find and graph the cumulative distribution function of X. Generate 1000 values for X and make an empirical probability histogram of the data. The random