If X is a binomial random variable, for what value of θ is the probability b(x;n,θ) a maximum?
Answer to relevant QuestionsIn the proof of Theorem 5.2 we determined the quantity E[X(X – 1)], called the second factorial moment. In general, the rth factorial moment of X is given by µ'(r) = E[ X(X – 1)(X – 2) · . . . · (X – r + ...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 – ...If X is a random variable having a geometric distribution, show that P(X = x + n| X > n) = P(X = x) Approximate the binomial probability b(3; 100, 0.10) by using (a) The formula for the binomial distribution and logarithms; (b) Table II. Use the result of Exercise 5.3 to show that for the Bernoulli distribution, (a) α3 = 1 – 2θ / √(1 - θ), where α3 is the measure of skewness defined in Exercise 4.26 on page 129; (b) α4 = 1 – 3θ(1 - θ) / √(1 - ...
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