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Unformatted text preview: STAT 333 Fall 2010 October 18, 2010 [Thursday, Sep 30, 2010] Objective: Conditional expectation and double-average theorem Example 1 Consider i.i.d. Bernoulli trials with p r ( S ) = p, < p < 1. Define that T s = the waiting time for the first S , and T ss = the waiting time for the first SS in a row. Then T s Geo ( p ). Find E( T s ) and E( T SS ). Example 2 A miner is trapped in a mine. There are three doors in front of the miner. The first door leads to a tunnel that takes him to safety after two hours of travel. The second door leads to a tunnel that returns him to the mine after three hours of travel. The third door leads to a tunnel that returns him to his mine after five hours. Assuming that the miner is at all times equally likely to choose any one of the doors. Let X = the number of hours to reach the safe place. 1. Find E(X) and Var(X). 2. If the first door leads the miner to the safe place after n 1 hours, and the other two doors lead the miner to the safe place after n 2 and n 3 hours, respectively. Find E(X) and Var(X). Example 3 Let N be the number of total claims to an insurance company in 2011. Sup- pose that N follows a Poisson distribution with rate , such that N Poi ( ). Denote X i ,i = 1 , 2 ,... the amount payed to the i-th claim. We assume that N and X i ,i = 1 , 2 ,......
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