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Question: a probability generating function encodes the probabilities of different events...

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A probability generating function encodes the probabilities of different events happening with a gener ating function. For example, consider tossing a coin until the first time you get a heads. Then, let pn be the probability that you had to flip the coin n times until you got a heads for the first time. The probability generating function for this problem would be n-1 (a) If your coin has a probability p of landing heads on any particular coin toss, find the value of pn in terms of p. (b) Find a closed form for the probability generating function for this coin. (c) What is f(1)? Give a non-algebraic reason for why this is true (d) The average number of times you need to flip the coin in order to get a heads would be given by n Pn n-1 Compute this value using your generating function from part (b)

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