Every morning, I send a random number of emails to my friends. The
number of emails I send every day follows an independent Poisson distribution with mean µ. For every email that I send, I get a reply before the next morning with probability p. Every email that isn’t replied to remains in the person’s inbox and will be replied to with probability p every day until it eventually is replied to. Once an email is replied to, it is deleted. Let Xi be the total number of replies I have gotten by the ith day just before I send out any new emails. In particular X1 = 0 and X2 is the number of immediate replies from the first batch of emails I send. All replies are independent. What is the distribution of Xn? State carefully any results or assumptions that you are making.
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