STAT 36-217, HW 5, due Thursday 10/8/2009, 10:30 AM1. The joint PMF ofXandYare given byp(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.CalculateE[X|Y=i] fori= 1,2,3. AreXandYindependent?2. IfXandYare independent Poisson distribution having the same parameter 1. Cal-culate the conditional distribution ofX=kgivenX+Y=n.3. LetX1andX2be independent geometric random variables having the same parameterp. CalculateP(X1=i|X1+X2=n). (Hint: you can first guess the result and theverify the answer. The idea is a follows. Suppose a coin having probabilitypof comingup heads is continuously flipped. If a second hear occurs on flip numbern, what is theconditional probability that the first head was on flip numberi,i= 1,2, . . . , n-1?).4. There arencomponents. On a rainy day, componentiwill function with probabilitypi; on a non rainy day, componentiwill function with probabilityqi, fori= 1,2, . . . , n.It will rain tomorrow with probabilityα.Calculate the conditional expected num-ber of components that function tomorrow, given that it rains. Also, calculate theexpected number of components that function tomorrow.
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