Problems in Probability

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Fall 2011 Homework 10: Question 1
Consider the following:
Judit plays in a total of N ~ Geom(s) chess tournaments in her career. Suppose that in each tournament she has probability p of winning the tournament, independently. Let T be the number of tournaments she wins in her career.
(a) Find the mean and variance of T.
(b) Find the MGF of T. What is the name of this distribution (with its parameters)?
Solution: (a) By Adam's Law, E(T) = E(E(T|N)) = E(Np) = p(1 - s)/s. By Eve's Law, Var(T) = p(1 - s)(s + (1 - s)p)/s^2. (b) From the MGF T ~ Geom(theta), with theta = s/(s+(1-s)p). The distribution of T can also be obtained by a story proof. Consult iTunes course for full solution.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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