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Unformatted text preview: today with probability 0.4; and in any other case the weather today will, with probability 0.8, be the same as the weather yesterday. Determine P for this Markov chain. 3. (30 points) Let X and Y be two discrete random variables with joint density function given by y 1 2 3 1 1 12 1 6 1 12 x 2 1 6 1 4 1 12 3 1 12 1 12 Compute the probability of the following events (a) (5 points) X is less than 2.5. (b) (5 points) X is even. (c) (10 points) XY is even. (d) (10 points) Y is odd, given that X is odd. 1 4. (30 points) The joint probability mass function of X and Y , p ( x, y ), is 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 (a) (20 points) Compute E [ X  Y = i ] for i = 1 , 2 , 3. (b) (10 points) Are the random variables X and Y independent? Explain your answer. 2...
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 Spring '08
 Dolinskaya

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