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.