Let X be a discrete random variable with pmf (a) Find the pmf for Y = X². (b) Find the pmf for U = X + 2. f(x) = px(x) = 1/8, 1/4, 1/4, 1/4, 1/8, 0, x 2 x = -1 x = 0 x = 1 x = 2 otherwise

The probability mass function for a discrete random variable X is defined as ((1+0)-(x) 0; x = 0,1,2,3,..., η (x) = {(1+0) (2) *; fx(x) 0; e.w. where 0 > 0. Show that it is probability mass function. Find its mean and variance.

Let X and Y be two random variables with Var(X) = 6, Var(Y ) = 3, andthe correlation ρ(X, Y ) = −1/4. Find the value of Var(X − 2Y + 7).

E(XY) – E(X)E(Y) 1. The correlation of X and Y is defined as px,y if X and Y are independent, then px,y = 0. Quickly show that o(X)o(Y) 2. Let X be a random variable with P(X = k) = | for k = -1,0, 1. Let Y = X². (a) What is the p.m.f. of Y? (b) Are X and Y independent? (c) Calculate E(XY) – E(X)E(Y). (d) True or false (for any random variables X and Y): X and Y are independent if and only if E(XY) – E(X)E(Y). 3. An exam consists of twenty multiple choice questions. Each question has five different possible answers, only one of which is correct. A person randomly guesses on each question. What is the variance of the number of questions that are correct? 4. Suppose ten people get on an elevator. There are eight buttons representing floors on this elevator. Each person presses a random button. The elevator will only stop at a floor if at least one person pressed that floor's button. What is probability the elevator stops at exactly six floors? What is the variance of the number of floors…