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131A_1_PracticeFinal_Sol

131A_1_PracticeFinal_Sol - EE 131A Probability Instructor...

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Unformatted text preview: EE 131A Probability Instructor: Lara Dolecek Fall 2010 Practice Final December 6, 2010 Maximum score is 200 points. You have 180 minutes to complete the exam. Please ShOW your work. Good luck! Problem Score Possible 1 l 15 2 10 3 20 4 15 5 10 6 r 15 7 l F 15 8 20 9 20 10 ‘ 25 11 ’ "A 10 12 25 Total 200 1. (15 pts) The random variable X has Cdf as shown in Figure. -1 1/2 0 1 Compute (a) P1X < _11a (b) P[X g —1],and (c) PHX — 0.5] > 0.5]. 2. (10 pts) Suppose ( X , K Z) are distributed as -ollows: 0 0) with probability 0.25, 5 5) with probability 0.257 ,5, 0) with probability 0.25, 0 5) with probability 0.25. (XMZ): (a) Are X and Y independent ? (b) Are X, Y and Z independent ? 5W ' if; i It, A 5 ‘ xi _ if ’ 5 f 27 ‘ 3 §"' ‘ g k :13: g i ,i 3 E i , 2 e a {2 c’ E V” i 7" 3 x’ -\ x A. : 3 ~:5 0 l: a 3 - A‘ $1 E § 3; x—a \ w 9.. CD 3 (20 pts) Suppose X17 X23 7 “n are ' pende_1t an" uniformly disjributed on {a b] Let Y = maX{X1,X2, . . . ,Xn}. (a) Find pdf of Y. (b) What are E[Y] and VAR(Y)? 4: (15 pts) Fifty—two percent of the residents of New York are in raver of outlawing cigarette smoking in publicly owned places. A pollster randomly picks n people and determines what percentage of them are in favor of banning smoking. How large should n be such that with at least 95% probability, greater than 50 percent of people polled would favor banning smoking in public places? Hint: Use central limit theorem. Also <1>(i.645) = 0.95. 5; (10 pts) Suppose X and Y are random variables each with mean 0 and variance 1, and with the correlation coefficient pr. Show that (X —~ pX YY) and Y are uncorrelated, 7 and that (X — prY) has mean 0 and variance 1 - pimp 6. ( 15 pts) Supnose X 1 Gaussian with mean m. and variance (1?. (21) Find pdf of Y = |X| (b) Find E[Y] OK to express in terms of <I>(-), WhCI’C <I>(z) standard normal. @0me w v (15 pts) Suppose X is Geowletric random variable With parameter p. (a) Compute the characteristic function of X , (I)X(w). (b) Compute E[X] using (I>X(w). WM, 4 , z 8. ( 20 pts) Let X1 be Chosen uniformly on (0,1), X2 be chosen uniformly on (0, X1) and X3 be Chosen uniformly on (0, X2). (a) Find the joint pdf of X1, X2 and X3. (b) Find the marginal pdf of X 3. 9. ( 20 pts) Consider random variables X and Y jointly distributed according to the joint density function f (x, y) = ke’(m2+$y+y2)/2 over —oo < m, y < 00 Where the constant k is chosen to make f ((L‘, y) a valid joint pdf. (a) Find the constant k. (b) Find fX(a:). (0) Are X and Y independent ? dywmm ‘ 10 10. (25 pts) Suppose X and Y are jointly Gaussian with E [X i = 0 an covariance matrix 1 0 KXY = [ J O 4 (a) What is fy(y) ? (b) Suppose Z 2 X — Y. What is fz(z)? (0) Suppose also W 2: X —|— Y. What is fan/(2,10) ? (d) What is fX,Z(X, Z) 7 xx. 11 11. (10 pts) Let X1, X2, X3... be independent and identically distributed with X1 = 0 with probabilityp and X1 = 1 with probability l—p. Calculate limrH00 ((X1)2 + (X2)2 + - - ' —l— (Xn)2) fly; :N 1 3 i a. i I \f‘f ’ ‘ . E K?) ‘ .1 ‘ r 5 .a a 4‘" g k 5 ~ 5 3’ ~ i\ g ,3 i t 1, i .2 ‘ u x E ‘ 3 5 g ‘\ E g ( if g .P g E 12 12. ( 25 pts) Suppose there are d distinct cards in the deck, labeled 1 through d. Draw cards with replacement until exactly k (k: g d) distinct cards have been obtained, Let Sh denote the total number of cards drawn. Compute E [Sh] Hint: Use Xi = 81H — Si and express Sk in terms of X1, X2, 13 ...
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