Let X1, , Xn be a sample from a normal distribution with 0 mean, and unknown variance = ^2. Assume the prior density of is p() = e^(), gt; 0.

Let X1, · · · , Xn be a sample from a normal distribution with 0 mean, and unknown variance θ = σ^2. Assume the prior density of θ is p(θ) = θe^(−θ), θ > 0. Determine the Bayesian estimator of the unknown variance given by the mode of the posterior distribution.