Statistics. Probability Mass Function. Expected Value.

MATH/STAT 416 HW 3 Due Fri Sept 18 (1) A bowl contains 10 each of red, blue, and yellow marbles. Three marbles are chosen randomly without replacement. Let R be the number of chosen red marbles, and let N be the number of chosen marbles which are not red. (a) Write down the probability mass function for R. (b) Write down the probability mass function for N. (2) A card is drawn at random from a standard deck of cards. If it's an Ace you win $5. If it's a J,Q,K, you have to pay $5. If it is a numbered card (2,3,. ..10), you win the value of the card in dollars. What is the expected value of your winnings? (3) Suppose that a biased coin has probability of heads equal to p. Find the expectation of the number of heads minus the number of tails for n inde- pendent flips. (4) Consider the sample space S = {1, 2, 3, .. . 100}. Let X(n) = n. (a) Compute E(X) for the uniform distribution. (b) Compute E(X) for the probability distribution P(n) = 150 if n even 150 if n odd (5) Let S = {1, 2, 3, ...} with the probability distribution P(n) = n(n + 1) Find the expectation of X(n) = 10 if it's defined