Use the moment–generating function derived in Exercise 5.20 to show that for the geometric distribution, µ = 1/θ and σ2 = 1 – θ / θ2.
Answer to relevant QuestionsDifferentiating with respect to θ the expressions on both sides of the equation Show that the mean of the geometric distribution is given by µ = 1/θ. Then, differentiating again with respect to θ, show that µ'2 = 2 – ...Verify the expression given for E[X(X – 1)] in the proof of Theorem 5.7. Derive the formulas for the mean and the variance of the Poisson distribution by first evaluating E(X) and E[ X(X – 1)]. In a certain city, incompatibility is given as the legal reason in 70 percent of all divorce cases. Find the probability that five of the next six divorce cases filed in this city will claim incompatibility as the reason, ...A manufacturer claims that at most 5 percent of the time a given product will sustain fewer than 1,000 hours of operation before requiring service. Twenty products were selected at random from the production line and tested. ...
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