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A-random-variable-X-has-a-Gamma-distribution-with-parameters-and-gt-0-The-p-d-f-has-the-form-f-x-1-x-n-1-e-x-for-x-gt-0-1-0-x-1-e-x-dx-a

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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.
Arandomvariable,X,hasaGammadistributionwithparametersαandβ,(α,β>0).Thep.d.fhastheformf(x)=1Γ(α)βαxn−1e−x/β,forx>0,Γ(α)=1βα∫0∞xα−1e−xdxa∖ShowthattheGammadensityisaproperp.d.f.b∖Findthemean,variance,andmoment−generatingfunctionoftheGammadistribution.c∖Findthefourthmomentusingthedefinitionofmoments.

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