The negative binomial distribution is sometimes defined in a different way as the distribution of the number of failures that precede the kth success. If the kth success occurs on the xth trial, it must be preceded by x – k failures. Thus, find the distribution of Y = X – k, where X has the distribution of Definition 5.4.
Answer to relevant QuestionsWith reference to Exercise 5.16, find expressions for µY and σ2Y. In exercise The negative binomial distribution is sometimes defined in a different way as the distribution of the number of failures that precede the kth ...Differentiating 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 – ...When calculating all the values of a Poisson distribution, the work can often be simplified by first calculating p(0;λ) and then using the recursion formula Verify this formula and use it and e–2 = 0.1353 to verify the ...Differentiating with respect to λ the expressions on both sides of the equation Derive the following recursion formula for the moments about the mean of the Poisson distribution: For r = 1, 2, 3, . . .. Also, use this ...With reference to Exercise 5.45 and the computer printout of Figure 5.1, find the probability that among 10 cars stolen in the given city anywhere from 3 to 5 will be recovered, using (a) The values in the P(X = K) column; ...
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