This is the question: (I am doing currently the a) )
https://i.stack.imgur.com/82F7X.png
So far, I say that:
$\text{Posterior probability} \propto \text{Likelihood}\times \text{Prior probability}.$
Then, we have Prior already, so we just need to compute Likelihood. I assume that samples are independent, so
$f_{X1,X2,X3,X4}(x_1,x_2,x_3,x_3\mid \theta) = \frac{1}{\theta} \times\frac{1}{\theta} \times\frac{1}{\theta} \times\frac{1}{\theta} = \frac{1}{\theta^{4}}$.
Therefore, $\text{Posterior probability} \propto \frac{1}{\theta^{4}}\times \frac{1}{10}.$
I am not sure how to derive the expression from the question:
$\pi(\theta | D) = c \cdot \frac{1}{\theta^{4}} , 8<=\theta<10.$
I know that $\pi$ should integrate to 1 to be a valid density.
So, I say that $\int_{8}^{10} c \frac{0.1}{\theta^{4}} d\theta= 1. $
And I find that c=31475.44.