I would be very grateful to get some help with the following problem.
Suppose $X_1, X_2, · · · , X_n ∼ N(0, θ)$ where $θ ∼ Gamma(3, 0.5)$, that $$p(θ) = \frac{θ^2e^{-θ\over 2}}{2^3Γ(3)}\ , θ > 0.$$
Find the posterior distribution that $p(θ|x_1, x_2, · · · , x_n)$, subject to a constant $c_p$.
I know that in Bayesian statistics, we always have to calculate the posterior dist., where $$π(θ|data) = π(θ) ∗ p(data|θ)/p(data).$$
And the posterior density of θ can be viewed as $$c_N f_1(θ), where f_1(θ) = π(θ) ∗ p(data|θ),$$ and $c_N$ is the normalizing constant.
But i am not sure how to approach this problem to get the posterior distribution as required, and would appreciate any help. Thanks!
You can look up a standard table (eg https://en.m.wikipedia.org/wiki/Conjugate_prior). In this case if you have a Normal with known mean (zero mean) and a Gamma distributed precision parameter then the prior is conjugate. As written, it’s not. Ie you need the inverse of the variance to be distributed Gamma and not the variance.