I'm getting stuck on this problem that I'm looking at and I am unable to figure out how to start solving this:
"A manufacturing process produces a large number of components each day, and most of these components will meet all specifications listed for the components. Let $p$ be the probability that a component will meet all specifications, and assume that one component meeting specifications is independent of other components meeting specifications. The value of $p$ varies from day to day according to the prior distribution:
$$ \pi(p) = 9p^8\quad 0\le p\le 1 $$
A sample of $n = 5$ components from the production of one day was inspected, and it was found that 4 of the 5 components met all specifications. Find the Bayes estimator of $p$ for this day."
I'm assuming I may to use Baye's Theorem in conjunction with a point estimator, but I'm not entirely sure how to set it up...
Thanks in advance for any input!
I will construe your statement about the prior distribution to mean the prior density.
The likelihood is $$ L(p) = \Pr(\text{4 successes in 5 trials} \mid p) = \binom 5 4 p^4(1-p)^1. $$ We don't care about the constant $\dbinom 5 4$ because we will normalize at the end. So $L(p)\propto p^4 (1-p)$ and the prior density is proportional to $p^8$. Multiply these, getting $p^{12}(1-p)$. If you want the posterior density, you need a constant $c$ such that $$ \int_0^1 p^{12}(1-p)\,dp = 1. $$ But you should also recognize this as the Beta density with parameters $13$ and $2$.