Question: The factorial moment generating function of a discrete random variable X
The factorial moment-generating function of a discrete random variable X is given by
Show that the rth derivative of FX(t) with respect to t at t = 1 is µ'(r), the rth factorial moment defined in Exercise 5.11.
Answer to relevant QuestionsWith reference to Exercise 5.12, find the factorial moment–generating function of (a) The Bernoulli distribution and show that µ'(1) = . and µ'(r) = 0 for r > 1; (b) The binomial distribution and use it to find µ and ...Prove Theorem 5.5. Theorem 5.5 b*(x;k,θ) = k/x. b(k;x,θ) A variation of the binomial distribution arises when the n trials are all independent, but the probability of a success on the ith trial is θi, and these probabilities are not all equal. If X is the number of successes ...Use repeated integration by parts to show that This result is important because values of the distribution function of a Poisson random variable may thus be obtained by referring to a table of incomplete gamma functions. 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, ...
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