MA519_JAN08

# MA519_JAN08 - minutes Thus the birth minutes of diﬀerent...

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QUALIFYING EXAMINATION January 2008 MA 519 - Prof. Sellke 1. A box contains 2 red, 3 white, and 4 black beads. A blind person randonly arranges the 9 beads on a circular loop of string. What is the probability that the beads are grouped together by color (with the red beads together in a group, the white beads together, and the black beads together)? 2. A fair, six–sided die is rolled repeatedly until all six possible results have been obtained. Let X = the number of rolls to see the ﬁrst 1 . Y = the number of rolls to see the ﬁrst 6 . So, if the successive results are 2556431, then Y =4and X =7.Find E { X | Y =5 } . 3. Let’s ignore leap years and assume that there are 60 × 24 × 365 = 525 , 600 minutes in a year. Let’s also assume that each person is equally likely to have been born in any of these 525,600 minutes, with diﬀerent people having independent birth
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Unformatted text preview: minutes. Thus, the birth minutes of diﬀerent people are independent discrete–uniform random variables on the set { 1 , 2 , 3 ,... , 525 , 600 } . About how many people do you need in a room to have a 50–50 chance (that is, probability 1 2 ) of there being at least one birth minute shared by two or more people? 4. Let X and Y be independent U [0 , 1] random variables. Find the density of Z =-Y/ ln( X ). 5. Let N be geometric ( p ), with P { N = k } = (1-p ) k-1 p, k = 1 , 2 ,... , EN = 1 p , var( N ) = q p 2 . Let X 1 ,X 2 ,... be independent unit–exponential random variables, with density f ( x ) = e-x , x > 0, and EX n = var( X n ) = 1. The X n ’s are independent of N . Let Y = N X i =1 X i . Find the variance of Y ....
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