1. Consider an experiment that consists of tossing a die and flipping a coin at the same time. Define the following random variables: X1: The number of dots appearing on the die. X2: The sum of the number of dots on the die and the indicator for the coin, where the value of "Head" is 1 and the value of a "Tail" is zero. a. How many possible outcomes are there, and what are they? b. What are the possible values and associated probabilities for X2? What is the probability P(X1=3, X2=3)?2. Let Y be a random variable with p(y) given in the accompanying table. Find E (Y), E (1 / Y), E (Y2 1), Var(Y), and Sthev(Y). 3. X and Y are discrete random variables with the following joint distribution: Value of Y 14 22 30 40 65 1 0.02 0.05 0.10 0.03 0.01 Value of X 5 0.17 0.15 0.05 0.02 0.01 10 0.02 0.03 0.17 0.08 0.09 That is, Pr (X = 1, Y = 14) = 0.02, and so forth. a. Calculate the probability distribution, mean, and variance of Y. b. Calculate the probability distribution, mean, and variance of Y given X=10. c. Calculate the covariance and correlation between X and Y.4. Compute the following probabilities (hint: you may find these tables useful): a. If Y is distributed t10, find Pr (Y > 1.75). b. If Y is distributed too, find Pr (-1.99