1. ## Bayesian Methods

Hey guys,

I'm doing revisions for Statistics and came across this question that I can't tackle. Bayesian is the worst topic for me because I can't seem to understand it. I hope you guys can help me out!

Records collected over many years show that on average 74% of first year students have IQ's of at least 155. Of course, the percentage 100 x theta varies from year to year and this variation is measured by a standard deviation of 3%. Suppose that the proportion of theta has a beta distribution. A sample of 30 first year students enrolling in 2007 showed that only 18 of them have IQ's of at least 115.
What is the prior estimate of theta in 2007?
What is the maximum likelihood estimate of theta in 2007?
Determine the mean of the posterior distribution of theta in 2007.

2. Let X be the number of people achieving the stated IQ. We are given that $\displaystyle \theta \sim Beta(\alpha, \beta)$ and clearly $\displaystyle X|\theta \sim Bin(30, \theta)$. First, you need to solve for $\displaystyle \alpha, \beta$ using the information given (you know the moments, so you can recover these parameters).
Getting the prior estimate is trivial; it's just .74. The MLE for theta in 2007 is also trivial; just the sample proportion. To get the posterior mean you need the posterior distribution. Use the fact that the posterior is proportional (treating X as a constant) to the joint distribution to get that. It happens to be the case that the posterior is $\displaystyle Beta(X + \alpha, n - X + \beta)$, but you should make sure you can do it working with the densities. Answering the last question just requires knowing the mean of that.