A random variable, X, has a Gamma distribution with parameters α and β, (α, β>0). The p.d.f has the form f(x)=(1/(Γ(α)β^α ))x^(n−1) e^(−x/β) , for x>0 , Γ(α)=(1/β^α )∫_0 ^∞ x^(α−1) e^(−x) dx a\ Show that the Gamma density is a proper p.d.f. b\Find the mean, variance, and moment-generating function of the Gamma distribution. c\Find the fourth moment using the definition of moments.


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Question Number 102080 by Ar Brandon last updated on 06/Jul/20
$A random variable, X, has a Gamma distribution with parameters α and β, (α, β>0). The p.d.f has the form f(x)=(1/(Γ(α)β^α ))x^(n−1) e^(−x/β) , for x>0 , Γ(α)=(1/β^α )∫_0 ^∞ x^(α−1) e^(−x) dx a\ Show that the Gamma density is a proper p.d.f. b\Find the mean, variance, and moment-generating function of the Gamma distribution. c\Find the fourth moment using the definition of moments.$
$A random variable, X, has a Gamma distribution with$ $parameters α and β, (α, β > 0) . The p . d . f has the form$ $f (x) = \frac{1}{Γ (α) β^{α}} x^{n - 1} e^{- x / β}, for x > 0, Γ (α) = \frac{1}{β^{α}} \int_{0}^{\infty} x^{α - 1} e^{- x} dx$ $a ∖ Show that the Gamma density is a proper p . d . f .$ $b ∖ Find the mean, variance, and moment - generating function of$ $the Gamma distribution .$ $c ∖ Find the fourth moment using the definition of moments .$
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