507-F04 - case that all three get heads or tails they...

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Common Quals-- Probability Questions for OR and Stat Tracks, Fall 2004 DO THREE OUT OF FOUR QUESTIONS (If you do all four, only the first three problems will be graded) 1. A stick of length 1 is cut into two pieces at a randomly chosen point uniformly distributed along its length. Find the mean and the standard deviation of the smaller piece. 2. The conditional covariance of X and Y , given Z , is defined by ]. | ]) | [ ])( | ([ [( ) | , cov( Z Z Y E Y Z X E X E Z Y X = Show that ]). | [ ], | [ cov( )] | , [cov( ) , cov( Z Y E Z X E Z Y X E Y X + = 3. A father asks his three children to cut their backyard lawn. Since he does not specify which of the three is to do the job, each tosses a coin to determine the odd* person, who must then cut the lawn. In the
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Unformatted text preview: case that all three get heads or tails, they continue tossing until they reach a decision. Let p be the probability of heads and q =1-p , the probability of tails. Find the probability that they reach a decision in less than n tosses. If p =1/2, what is the minimum number of tosses required to reach a decision with probability 0.95? *If only one child gets a head (or tail), this child is “odd.” 4. Under what conditions is the sum of two independent binominal random variables, with parameters (n,p) and (m,q) respectively, a binomial random variable? Prove your answer....
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