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131A_1_MidtermInClassSolutions

# 131A_1_MidtermInClassSolutions - EB 131A Probability...

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Unformatted text preview: EB 131A Probability Instructor: Lara Dolecek Fall 2010 Midterm Wednesday, October 27., 2010 Maximum score is 100 points. You have 110 minutes to complete the exam. Please show your work. Good luck! Your Name: Your ID Number: Name of person on yOur left: Name of person on yOur right: I Problem Score Possible 1 6 ""‘ 2 15 3 8 4 12 5 15 6 8 " 7 *“m 8 “l 8 "j: 10 : 9 18 ” Total 100 1. (6 pts) Suppose 10(14): 1/3, P(AU B) 1 1/2 and PU—lﬂ B] m 1/5. Find 13(3). A s “4: 4 2. (4+5+6 pts) Let X and Y be independent and uniform on {1, 2, . . . 1M}. ‘3‘ (a) P(X = Y). (b) P(X 3 Y). (c) pmf of U Where U = iX -— Y]. \ 3 ,2 3 3. (4+4 pts) Suppose X and Y are independent random variables With ﬁnite ﬁrst and. second moments. Let Z 2 3X + 5)”. Compute mean and variance of Z in terms of E(X),E(Y), VARQC) and VAR(Y). 5;. v ‘2; x ’K »‘ Q: \3 A 4%. (12 pts) A 52mcard deck consists of 4 suits (clubs, (iiamonds, hearts, Spades), and each suit has 13 cards (2, 3, 4, 5, 6, 7, 8, 9, 10, 5, Q, K, A). Suppose you are dealt a, poker hand consisting of 5 cards. Compute the probability of getting Bush. Flush hand is 5 cards of the same suit. ‘jmgmjf p j 4,} *' b 5 5. (15 pts) True or False. Circling the correct answer is worth +3 points, circling the incorrect answer is worth —u~1 points. Not circling either is worth 0 points. (a) Suppose X N NH), 1). Then P(X K {3) m 0.5. TRUE {/FALSE‘g k». N W/ (b) If A and B are mutually exclusive events, P(AUB) m: P(A) + PUB) —P(A)P(B). TRUE {FALSE} ((3) E[(X— mﬂ 1 13pm ~— m? Where m 2 E[X}. r”/‘I‘RUE/ FALSE (d) VAR(X + C) 2-": VARQQ + c (where c is a constant). TRUE (FALSE: (8) mmx ‘ Y) : VAR(X) w- VAMY) TRUE FALSE 6. (8 pts) Suppose that in a certain City 60% of residents are male and 40% Of residents are femaie. Suppose aiso that 50% of males and 30% of females cheer for Lakers. Find the probability that a Lakers fan is femaie. \ g1 f“. . 5" f ‘ .. gm I» {J [jw v ; (g .~ g «:3 7. (4+4 pts) Suppose X N NHL, 02). (54) Find PQX —-- 44% S a) in terms of Q function. (b) Find constants c and b such that Z = a + 14X is N(O, 1). ‘ f? 44 €444; 44’14: 4! 4 4 443 4:44 :4 4‘4” '44 ~44. 4.44 44' f 4 4 8. (8+2 pts) Let X be uniformly distributed on (01 1). Find cdf and pdf of Y m —% 111(1 ~ X ), where a > 0. “What is the name of this random variable ? Mew, k x A W '2... a i we (:— 3:13. W C r” v {2% «m , ",1 9. (4+6+8 pts) Suppose n bails are distributed in r boxes. Let Xi = 1% if box i is empty, and 0 otherwise, fer 1 g 2' g 'r (a) Compute E(X¢) for a ﬁxed 2', 1 g it 3 r. (b) Compute E(X1X?) for ﬁxed m, where 1 g 7353'“ g T and 25 % j. (e) Let 3 denote the number of empty boxes. Compute mean and variance of S. rx I‘- ““““ ‘ n 5/!2 : g G j ...... I :{J ) 11:2: ifs, :3“ E we"; :., {fl f: f :2 1.4.. {y‘LJML ‘ f1 III ‘ a ..._ \ u , ‘ M} ...
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131A_1_MidtermInClassSolutions - EB 131A Probability...

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