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1)An urn contains 10 white, 15 black and 20 red balls

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1)An urn contains 10 white, 15 black and 20 red balls. (a) (8pts) If two balls are randomly withdrawn without replacement, what is the probability that they are the same color? (b) (8pts) Suppose that you are randomly selecting balls with replacement, until a red one is obtained. What is the probability that at most 3 draws are needed? (c) (9pts) Suppose that you pick 1 ball and throw it away. Then pick a second ball and you see that it is black. What is the probability that the first ball was red? 2 2. Two fair 4-sided dice (tetrahedrons with four triangular faces labeled 1,2,3 and 4) are thrown. We observe the numbers on the bottom faces and denote by X the largest one. For instance, if one die has the bottom number 4 and the other 2, then X = 4, while if both bottom numbers are 3, then X = 3. (a) (8pts) Find P(X < 3). (b) (8pts) Compute E(X). (c) (9pts) Find the variance of the random variable Z = 2X − 4. 3 3. Let X have density function f (x) = c(3 − |x|), for x ∈ (−3, 3) [and 0 otherwise]. Find (a) (8pts) the constant c. (b) (8pts) the conditional probability P(X ≥ −2|X < 0) (c) (9pts) the density function of the random variable Y = |X|. 4 4. Determine whether the following statements are True or False. Justify your answer with a proof or a counterexample as appropriate. (a) (8pts). If A and B are disjoint events, then Ac and B are independent. (b) (8pts). If X ∼ N (1, 4), then P(X > −1) ≥ 1/2. (c) (9pts). For each p ∈ [0, 1], if X ∼ Ber(p) and Y = 1 − X, then Y ∼ Ber(p). 5 2 2. Urn U1 contains 3 red and 3 black balls, whereas urn U2 contains 4 red and 6 black balls. (a) If a ball is randomly selected from each urn, what is the probability that the two balls will be the same color? (b) If 2 balls are drawn without replacement from each urn, what is the probability that 1 red and 3 black balls are selected? (c) If a ball is drawn from U1 and put into U2 , and then a second ball is picked from U2 ; what is the probability that the second ball is red? 3 3. Three fair coins have their faces labeled with numbers 1 and 2 instead of heads and tails. Assume that you toss them and let X be the sum of the three numbers on top. For instance, X(111) = 3 and X(212) = 5. (a) Find P(X > 4). (b) Plot the p.m.f. of X. (c) Compute E(6 − X). 4 4. Determine whether the following statements are True or False. Justify your answer with a proof or a counterexample as appropriate. (a) If X ∼ P oiss(λ), then E(eX ) = eλ. (b) If Y is a random variable with P(Y = 0) = 0, then E(1/Y ) = 1/E(Y ). (c) If X and Y are Bin(n, p) random variables, then X + Y ∼ Bin(n, p). 5