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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