Show that the moment generating function of the negative binomial distribution is M(t) = pr[1 - (1 - p)et] - r. Find the mean and the variance of this distribution. Hint: in the summation representing M(t), make use of the MacLaurin's series for (1 - w) -r.
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The negative binomial distribution models the number of trials needed to achieve a fixed number of successes in a sequence of independent and identically distributed Bernoulli trials. If \( X \) is a negative binomial random variable with parameters \( r \) Show more…
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