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Theorem The limiting distribution of the gamma(α, β) distribution is the N (µ, σ 2 ) distribution where µ = αβ and σ 2 = α2 β. Proof Let the random variable X have the gamma(α, β) distribution with probability density function x β−1 e−x/α fX (x) = x > 0. αβ Γ (β) The moment generating function of X is MX (t) = (1 − αt)−β t< 1 . α The mean of X is E[X] = αβ and the variance of X is V [X] = α2 β. Subtract the mean and divide by the standard deviation before taking the limit. The transformation Y = g(X) = (X − αβ) . q α β =X . q α β − q β √ is a 1–1 transformation from X = {x | x > 0} to Y = {y | y > − β}. The moment generating function of Y is h MY (t) = E etY = i . √ √ α β − β t X E e . √ α β t X β E e = e−t √ = e−t √ = e −t β √ β . q α β MX t .q −β 1− t β t< q β. The limiting moment generating function of Y is lim MY (t) = lim e β→∞ β→∞ −t √ β .q −β 1− t β = et 2 /2 − ∞ < t < ∞, which is the moment generating function of a standard normal random variable. This limit can be found using the maple statement limit(exp(-t * sqrt(beta)) * (1 - (t / sqrt(beta))) ^ (-beta), beta = infinity); which confirms that the limiting distribution of the gamma distribution as β → ∞ is the normal distribution. 1