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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262—Discrete Stochastic Processes Problem Set #12 Issued: May 12, 2002 Not due Problem 1 Exercise 7.28 of the class notes. Problem 2 a Let X, Y, Z be random variables with densities f X , f Y , f Z . Show that E( X | Y ) = E(E( X | Y, Z ) | Y ). Assume that E( | X | | Y = y ) < . Note: This is referred to as the “tower property”, used in Lemma 7.4 among other places. It holds in full generality, regardless of whether the variables have densities or not. Though it’s an elementary result (and intuitively obvious to some extent), it is good to remind ourselves of why it holds true. b Show that every infinite subsequence of a martingale is a martingale, i.e. { Z n } n 1 is a martingale = ⇒ { Z n k } k 1 is a martingale for all n 1 n 2 . . . . Problem 3 A magician has a hat containing white rabbits and green rabbits. A rabbit is drawn from the hat, its color noted, and is then returned to the hat along with a new rabbit of the same color. Suppose that initially, the hat contains one rabbit of each color. Letting G n denote the number of green rabbits after
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