3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume...

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3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα, β) is αβ (a) Find the joint p.m.f. f(x,... ,n) for X,,-.. ,Xn (b) Show that the posterior distribution of A given (Xi, .,Xn) -(x1,,^n) is a gamma distribution with paranneters 1 + Ση! Χί and 1/(n +1) (c) Find the mean of the posterior distribution of .

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3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume...

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