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In an urn there are two balls, one with the number 2, and on

    In an urn there are two balls, one with the number 2, and one with the number 3. When you pickfrom the urn the chance of observing a ball with the number 2 on it is 1 , and the chance of observing2the number 3 is also 1. Now pick with replacement from the urn until the sum of the scores is 8 or2more. Let N be the number of balls chosen and M be the sum of the scores.(a) Write down the joint probability mass function for N and M.(b) Find the marginal probability mass function for M.(c) Find P(N = 4|M = 9).(d) Find E(M − 2N).We saw in class that a coin-flipping game that pays $2n if the first head appears on the nth (that is,a random variable X such that P (X = 2n) = 1/2n for n = 1, . . . , ∞) toss has infinite expectation.(a)

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