Suppose the distribution of y i | is Poisson, We will obtain a sample of observations, y i , . . . , y

Suppose the distribution of y_i| λ is Poisson,

We will obtain a sample of observations, y_i , . . . , y_n. Suppose our prior for λ is the inverted gamma, which will imply

a. Construct the likelihood function, p(y₁, . . . , y_n | λ).b. Construct the posterior density

c. Prove that the Bayesian estimator of λ is the posterior mean, E[λ | y₁ , . . . , y_n] = y̅.

d. Prove that the posterior variance is Var[λ | y_l, . . . , y_n] = y̅/n.

exp(-1)2% y!! exp(-1)a ( + 1) f(y; |2) = %3D 0, 1, ..., > 0.