91Ó°ÊÓ

The posterior distribution is given by Bayes' Rule, which is the product of the likelihood function and the prior distribution. In this case, we have the likelihood function as the pdf of uniform distribution, \(f(x;\theta) = \frac{1}{\theta}\) for \(0<x<\theta\), and the prior distribution of \(\theta\) as \(g(\theta) = \frac{2}{\theta^3}\) for \(1 < \theta < \infty\). This gives us the joint distribution, which upon normalizing gives us the posterior distribution. Considering our sample size of 4, the likelihood part equates to \((\frac{1}{\theta})^4\) for \(Y_4 < \theta\). Therefore the un-normalized posterior distribution \(h(\theta | Y_4)\) is \(\frac{2}{\theta^3} * (\frac{1}{\theta})^4\) if \(Y_4 < \theta\), and 0 elsewhere.

The next step is to normalize the computed posterior distribution, which entails integrating over all possible values of \(\theta\) and equating it to 1. Therefore, we need to solve: \(\int_{Y_4}^{\infty} h(\theta | Y_4) d\theta = 1\). This will provide us with a normalization constant that will make the posterior distribution a proper pdf.

Having the normalized posterior distribution, we find the Bayesian estimator \(\delta(Y_4)\) with the given loss function \(|\delta(y_4)-\theta|\). The Bayesian estimator minimizes the expected posterior loss and can be calculated by using this formula: \(E[L(\theta, \delta(Y_4))|Y_4]\). In our case, by applying the formula, we have the Bayesian Estimator to be the median of the posterior distribution.