Problems in Probability

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Fall 2011 Strategic Practice 9: Section 3 (Conditional Expectation) - Question 2
Give a short, direct proof that a Geom(p) r.v. has mean q/p, for q = 1 - p.
Solution: Let X ~ Geom(p), interpreted as the number of failures before the first success in independent Bern(p) trials. Let A be the event that the first trial is a success. E(X) = E(X|A)P(A) + E(X|Ac)P(Ac) = 0p + (1 + E(X))q, where E(X) = q/p.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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