Hi,

please see below my task and the answer I already got.

My problem is that I do not understand the context and I still have a lot of question marks.

Please explain the task, better to say the answer in all details (comments in bold) so that I'm able to follow. I'm really struggling with this topic and highly appreciate your help.

Task:

Let 1, 2, ..., 10 be a random sample from a gamma distribution with = 3 and = 1/. Suppose we believe that follows a gamma-distribution with = 17 and = 72 and suppose we have a trial (1 , ... , ) with an observed = 39.29

a) Find the posterior distribution of .

b) What is the Bayes point estimate of associated with the square-error loss function?

c) What is the Bayes point estimate of using the mode of the posterior distribution?

Answer I already got:

Step 1:

a) We have the prior distribution of as gamma with = 17 and = 72, and the likelihood function of the sample as the product of gamma densities with = 3 and = 1/:

Likelihood function: f(x, x, ..., x | ) = (1/(^3 * (3))^10) * exp(-sum(xi/)) Unclear how to get this liklihood function

Prior distribution: f() = (1/(^ * ())) * (^(-1)) * exp(-/)

The posterior distribution is proportional to the product of the likelihood and prior:

f( | x, x, ..., x) f(x, x, ..., x | ) * f()

f( | x, x, ..., x) (1/(^3 * (3))^10) * exp(-sum(xi/)) * (1/(^ * ())) * (^(-1)) * exp(-/) Unclear how to get from this line to the following line

f( | x, x, ..., x) (1/(^ * ())) * (^(+n-1)) * exp(-(1/)*(sum(xi)+1/)) Unclear how to get from this line to the following line / result, that ' = 20 and ' = 10/393.9.

This is the kernel of the gamma distribution with ' = + n and ' = 1/(1/ + (1/n)*sum(xi)), so the posterior distribution of is gamma with ' = 20 and ' = 10/393.9.

Step 2:

b) The Bayes point estimate of associated with the square-error loss function is the posterior mean of :

( | x, x, ..., x) = ' * ' = 20 * (10/393.9) = 0.506

Step 3:

c) The Bayes point estimate of using the mode of the posterior distribution is the mode of the gamma distribution with ' = 20 and ' = 10/393.9, which is ('-1)/' = 19.5/10.02 = 1.947.