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5. {15 Points] On average, how many friends do you need to have, in order to have a birthday party every
day? (Of course, in this problem we assume that the birthdays of all your friends are independently
and uniformly distributed over the 365 days of the year.) Hints: Assume that you have N friends. Arbitrarily put them in a single line. Let X1 be the birthday
of the ﬁrst person in line, X2 be the birthday of the second person in line, etc. Starting with the ﬁrst person, you ask them one-by—one for their birthday. Let W] be the number of people you need to ask until you hear a birthday you have not heard so far. Clearly,
W1 = 1. Starting with person W1 + 1, let W2 be the number of people you need to ask until you hear a
birthday you have not heard so far. 1rWhat is the distribution of W2? Starting 1with person W1 + W2 + 1, let W3 be the number of people you need to ask until you
hear a birthday you have not heard so far. 'What is the distribution of W3? Etc.
For this process to work, you need on average N friends, where N 3-3 E[W1 + W2 —|— - + - + W355]. n

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