Statistics

"To find the posterior distribution of θ in this Bayesian setting, we can apply Bayes' theorem. The posterior distribution is proportional to the product of the likelihood and the prior distribution. Given the information provided, let's calculate the posterior distribution step by step:

We are given:

Prior distribution of θ: θ ~ Gamma(ξ17, ξ18)

Likelihood of the data given θ: X1, X2, …, X10 ~ Gamma(3, 1/θ)

Observed sample mean: x̄ = ξ19

a) Find the posterior distribution of θ:

Using Bayes' theorem:

Posterior ∝ Likelihood × Prior

Posterior(θ | x1, ..., x10) ∝ [Π f(xi | θ)] * [Gamma(θ | ξ17, ξ18)]

Now, let's calculate the likelihood and prior.

Likelihood:

The likelihood is the product of the individual gamma distributions for each Xi:

Likelihood(θ | x1, ..., x10) ∝ Π (1/Γ(3)) * (θ^3) * (e^(-θxi))

Here, Π denotes the product over i from 1 to 10."