Assume our data Y = (y₁, y₂,…, y_n)^T given X is independently identically distributed, Y | X = x ^i.i.d. ∼ Exponential(λ = x), and we chose the prior to be X ∼ Gamma(α,β).

a. Find the likelihood of the function, L(Y;X) = f_{Y1,Y2,…,Yn|X}(y₁, y₂,…, y_n|x).

b. Using the likelihood function of the data, show that the posterior distribution is

c. Write out the PDF for the posterior distribution, f_X|Y(x|y).

d. Find mean and variance of the posterior distribution, E[X|Y] and Var(X|Y).

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Gamma(a + n,β + Σ=13).