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# Hw 3 - Stat 116 Homework 3 Solutions 1 Let A and B be the...

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Stat 116 Homework 3 Solutions April 16, 2008 1. Let A and B be the number of people on a given flight from airline A and B respectively. Notice the probability that any given person checks in is 9/10 so this random variable is distributed Bernoulli with 9/10 as the parameter. Therefore since A and B are the sums of such random variables the distribution for A and B must be binomial, i.e. P ( A = k ) = 10 k 9 10 k 1 10 10 - k , P ( B = k ) = 20 k 9 10 k 1 10 20 - k Then P (A is overbooked) = P ( A = 10) = 9 10 10 . = . 3486 P (B is overbooked) = P ( B 19) = 20 9 10 19 1 10 + 9 10 20 . = . 3917 So airline B have a greater chance of being oversold. 2. Let ξ i be the number of throws before ith distinct number to appear for the first time after i-1 distinct numbers has appeared. ξ i has a geometric distribution with success probability p i = 6 - i +1 6 , so E [ ξ i ] = 1 p i . ξ = 6 X i =1 ξ i E [ ξ ] = 6 X i =1 E [ ξ i ] = 6 X i =1 1 6 - i +1 6 = 14 . 7 1

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3. Let the number of signals received be ξ . P ( η = k ) = X n k P ( ξ = n ) P ( η = k | ξ = n ) = X n k λ n n !
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