Show that a gamma distribution with α > 1 has a relative maximum at x = β(α – 1). What happens when 0 < α < 1 and when α = 1?
Answer to relevant QuestionsExpand the moment–generating function of the gamma distribution as a binomial series, and read off the values of µ'1, µ'2, µ'3, and µ'4. Prove Theorem 6.1. Theorem 6.1 The mean and the variance of the uniform distribution are given by µ = α + β / 2 and σ2 = 1/12 (β – α)2 Show that the parameters of the beta distribution can be expressed as follows in terms of the mean and the variance of this distribution: (a) (b) If X is a random variable having the standard nor–mal distribution and Y = X2, show that cov(X, Y) = 0 even though X and Y are evidently not independent. If X and Y have the bivariate normal distribution with µ1 = 2, µ2 = 5, σ1 = 3, σ2 = 6, and ρ = 2/3 , find µY|1 and σY|1.
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