hw10 - X i = 1 if the i-th draw is an Ace = 0 otherwise...

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MATH 471 HW 10 Solutions Pengsheng Ji November 7, 2006 4.2.13 Observing tx - x 2 2 = - ( x - t ) 2 + t 2 / 2, we have Ee tX = Z e tx 1 2 π e - x 2 / 2 dx = e t 2 Z 1 2 π e - ( x - t ) 2 / 2 dx = e t 2 / 2 . 4.2.20 EXY = Z 1 0 Z 1 0 ( x 2 y + xy 2 ) dydx = 1 / 3 4.2.25 If X is uniform on (0,1) then EX = 1 / 2, but E (1 /X ) = Z 1 0 (1 /x ) dx = . Jensen’s inequality, see 4.3.23, implies that in general E (1 /X ) 1 /EX with equality if and only if X is constant. 4.3.7 Let
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Unformatted text preview: X i = 1 if the i-th draw is an Ace, = 0 otherwise. Then ∑ 13 i =1 X i is the number of Aces drawn, so EX = 13 EX i = 13 · (4 / 52) = 1. 4.3.17 N-1 is geometric(1/2) since it is the waiting time for a child different from the first one. EN=1+2=3....
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