Problem 19. (3.0 %o) Let X be a random variable whose transform is given by M_X(s) = (0.4 +0.6e^s)⁵⁰. Then,
(A) P(X =0) = P(X = 50)
(B) P(X = 51) ＞0
(C) P(X = 0)=0.4⁵⁰
(D) P(X= 50) = 0.6

Problem 4. (2.5 %) Which of the following statement is NOT true? (A) If A ⊂ B, then P(A）≤ P(B）. (B) If P(B）＞ 0, then P(A|B) ≥ P(A）. (C) P(A∩B) ≥ P(A）+P(B） - 1. (D) P(A∩Bc) = P(AUB) - P(B）.

Problem 5. (2.5 %) We toss two fair coins simultaneously and independently. If the outcomes of the two coins are the same, we win； otherwise we lose. Let A be the event that the first coin comes up heads, B be the event that the second coin comes up heads, and C be the event that we win. Which of the following statements is false? (A) Events A and B are independent. (B) Events A and C are not independent. (C) Events A and B are not conditionally independent given C. (D) The probability of winning is 1/2.

Problem 6. (2.5 %o) Suppose X, Y and Z are three independent discrete ran- dom variables. Then X and Y + Z are. _____ independent. (A) always(B) sometimes (C) never 

Problem 7. (2.5 %) Consider two random variables X and Y, each taking values in {1,2,3}. Let their joint PMF be such that for any 1 ≤ x,y ≤ 3, Then, (A) X and Y can be independent or dependent depending upon the strictly positive values (B) X and Y are always independent (C) X and Y can never be independent

Problem 8. (2.5 %o) We throw n identical balls into m urns at random, where each urn is equally likely and each throw is independent of any other throw. What is the probability that i-th urn is empty? (A) (B)(C) (D)

Problem 9. (2.5 %) For a biased coin, the probability of 'heads" is 1/3. Let h be the number of heads in five independent coin tosses. What is the probability P(first toss is a head | h = 1 or h = 5)?(A) (B)(C) (D)

Problem 10. (2.5 %) A well-shuffled deck of 52 cards is dealt evenly to two players (26 cards each). What is the probability that player 1 gets all the cases (A) (B)(C)(D)

Problem 11. (2.5 %) To obtain a driving license, John needs to pass his driving test. Every time John takes a driving test, with probability 1/2, he will clear the test independent of his past. John failed his first test. Given this, let Y be the additional number of tests John takes before obtaining a license. Then, (A) E[Y] =1(B) E[Y] =2(C) E[Y]=0

Problem 12. (25 %) Let Xi, 1 ≤ i ＜ 4 be independent Bernoulli random variable each with mean p = 0.1. Let X = That is, X is a Binomial random variable with parameters n = 4 and p = 0.1. Then, (A)E[X1 | X=2]=0.1(B)E[X1 | X= 2]=0.5(C)[X1 | X=2]= 0.25

Problem 13. (2.5 %o) Let X1, X2, X3 be independent random variables with the continuous distribution over [0,1]. Then P(X1＜X2＜X3)= (A) 1/6(B) 1/3(C) 1/2(D) 1/4

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