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# rec07 - E Y | X = 1(d Is there a choice for the unspeci±ed...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 7 September 30, 2010 1. Problem 2.35, page 130 in the text. Verify the expected value rule E [ g ( X,Y )] = summationdisplay x summationdisplay y g ( x,y ) p X,Y ( x,y ) , using the expected value rule for a function of a single random variable. Then, use the rule for the special case of a linear function, to verify the formula E [ aX + bY ] = a E [ X ] + b E [ Y ] , where a and b are given scalars. 2. Random variables X and Y can take any value in the set { 1 , 2 , 3 } . We are given the following information about their joint PMF, where the entries indicated by a * are left unspecified: 3 2 1 y 1/12 1/12 * 2/12 * 1 2 3 x 1/12 2/12 0 * (a) What is p X (1)? (b) Provide a clearly labeled sketch of the conditional PMF of Y given that
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Unformatted text preview: E [ Y | X = 1]? (d) Is there a choice for the unspeci±ed entries that would make X and Y independent? Let B be the event that X ≤ 2 and Y ≤ 2. We are told that conditioned on B , the random variables X and Y are independent. (e) What is p X,Y (2 , 2)? (If there is not enough information to determine the answer, say so.) (f) What is p X,Y | B (2 , 2 | B )? (If there is not enough information to determine the answer, say so.) 3. Problem 2.33, page 128 in the text. A coin that has probability of heads equal to p is tossed successively and independently until a head comes twice in a row or a tail comes twice in a row. Find the expected value of the number of tosses. Page 1 of 1...
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