The random variable X measures the concentration of ethanol in a chemical solution, and the random variable measures the acidity of the solution. They have a joint probability density function f(x, y) = A(20-x - 2y)

2.5.4 A fair coin is tossed four times, and the random variable X is the number of heads in the rst three tosses and the random variable Y is the number of bets in the last three tosses. (a) What is the joint probability mass function of X and Y? (b) What are the marginal probability mass functions of X and Y? (c) Are the random variables X and 1' independent? ((1) What are the expectations and variances of the random variables X and Y? (e) What is the covariance of X and Y? (1') If there is one head in the last three tosses, what is the conditional probability mass function of X '2' What are the conditional expectation and variance of X