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# HW05 - STAT 36-217 HW 5 due Thursday 10:30 AM 1 The joint...

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STAT 36-217, HW 5, due Thursday 10/8/2009, 10:30 AM 1. The joint PMF of X and Y are given by p (1 , 1) = 1 / 9 , p (2 , 1) = 1 / 3 , p (3 , 1) = 1 / 9 , p (1 , 2) = 1 / 9 , p (2 , 2) = 0 , p (3 , 2) = 1 / 18 , p (1 , 3) = 0 , p (2 , 3) = 1 / 6 , p (3 , 3) = 1 / 9. Calculate E [ X | Y = i ] for i = 1 , 2 , 3. Are X and Y independent? 2. If X and Y are independent Poisson distribution having the same parameter 1. Cal- culate the conditional distribution of X = k given X + Y = n . 3. Let X 1 and X 2 be independent geometric random variables having the same parameter p . Calculate P ( X 1 = i | X 1 + X 2 = n ). (Hint: you can first guess the result and the verify the answer. The idea is a follows. Suppose a coin having probability p of coming up heads is continuously flipped. If a second hear occurs on flip number n , what is the conditional probability that the first head was on flip number i , i = 1 , 2 , . . . , n - 1?). 4. There are n components. On a rainy day, component i will function with probability p i ; on a non rainy day, component i will function with probability q i , for i = 1 , 2 , . . . , n . It will rain tomorrow with probability α . Calculate the conditional expected num- ber of components that function tomorrow, given that it rains. Also, calculate the expected number of components that function tomorrow.
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