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Unformatted text preview: Midterm 1 Review 1. Suppose that a population consists of b black and r red balls. A sample of size k is drawn, without replacement, and X represents the number of red balls in the sample. (a) Using indicator variables, I 1 ,...,I k , which identify the colour of each ball in the sample, express X in terms of I 1 ,..., I k . (b) Using ordered arrangements of the balls, show that the probability that ball j in the sample is red is r/ ( r + b ) , j = 1 ,...,k . (c) Evaluate E ( X ) . (d) Using ordered arrangements of the balls, show that the joint probability that balls i and j in the sample are both red is r ( r 1) / { ( r + b )( r + b 1) } for i 6 = j . (e) Evaluate V ar ( X ) . Note: X has a hypergeometric distribution, but unless you prefer the direct approach, the solution outlined above is the simplest way to obtain E ( X ) and V ar ( X ) . 2. If X 1 ,X 2 ,... are i.i.d. with common p.m.f. P ( X = n ) = p (1 p ) n ( n = 0 , 1 ,... ), show that X 1  X 1 + X 2 = k U { , 1 , 2 ,...,k } (Discrete uniform over { , 1 , 2 ,...,k } ). 3. (Adapted from Chapter 3, Question 31 of Ross) Each outcome in a sequence of games is either a win with probability p or a loss with probability 1 p . A string of consecutive games with identical outcomes is called a run (or informally, a streak). So, for example, if the outcomes were W,W,L,W,W,W,L,... then the first run is of length 2, the second is of length 1, and the third is of length 3....
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 Fall '08
 Chisholm
 Probability

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