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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.

$$\mathrm{A}\:\mathrm{random}\:\mathrm{variable},\:\mathrm{X},\:\mathrm{has}\:\mathrm{a}\:\mathrm{Gamma}\:\mathrm{distribution}\:\mathrm{with} \\ $$ $$\mathrm{parameters}\:\alpha\:\mathrm{and}\:\beta,\:\left(\alpha,\:\beta>\mathrm{0}\right).\:\mathrm{The}\:\mathrm{p}.\mathrm{d}.\mathrm{f}\:\mathrm{has}\:\mathrm{the}\:\mathrm{form} \\ $$ $$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\Gamma\left(\alpha\right)\beta^{\alpha} }\mathrm{x}^{\mathrm{n}−\mathrm{1}} \mathrm{e}^{−\mathrm{x}/\beta} ,\:\mathrm{for}\:\mathrm{x}>\mathrm{0}\:\:,\:\:\Gamma\left(\alpha\right)=\frac{\mathrm{1}}{\beta^{\alpha} }\int_{\mathrm{0}} ^{\infty} \mathrm{x}^{\alpha−\mathrm{1}} \mathrm{e}^{−\mathrm{x}} \mathrm{dx} \\ $$ $$\mathrm{a}\backslash\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{Gamma}\:\mathrm{density}\:\mathrm{is}\:\mathrm{a}\:\mathrm{proper}\:\mathrm{p}.\mathrm{d}.\mathrm{f}. \\ $$ $$\mathrm{b}\backslash\mathrm{Find}\:\mathrm{the}\:\mathrm{mean},\:\mathrm{variance},\:\mathrm{and}\:\mathrm{moment}-\mathrm{generating}\:\mathrm{function}\:\mathrm{of} \\ $$ $$\mathrm{the}\:\mathrm{Gamma}\:\mathrm{distribution}. \\ $$ $$\mathrm{c}\backslash\mathrm{Find}\:\mathrm{the}\:\mathrm{fourth}\:\mathrm{moment}\:\mathrm{using}\:\mathrm{the}\:\mathrm{definition}\:\mathrm{of}\:\mathrm{moments}. \\ $$

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