Maximum Likelihood Function for uniformly distributed points I am currently trying to solve Exercise 22.10 from David MacKay's Book "Information theory, inference, and learning algorithms"

I have absolutely no idea on how to approach this task, and would highly appreciate if someone could point me to the right direction.
 A: Lets start with the one-dimensional case. The probability density function of the uniform distribution on [a,b] is $\frac{1}{b-a}\mathbf{I}_{[a,b]}(x)$.
The above formula also represents the likelihood of one point. What if you had multiple points (say, N)?
For a given sample of values, we can make a likelihood based on parameters $a$ and $b$ in the same way that you would for others:
$L(\mathbf{x};a,b)=\prod\limits_{i=1}^N \frac{1}{b-a}\mathbf{I}_{[a,b]}(x_i) = (\frac{1}{b-a})^N\prod\limits_{i=1}^N\mathbf{I}_{[a,b]}(x_i)$
This likelihood function has the following properties:
1) If any sample value falls outside [a,b], then the likelihood becomes zero.
2) The smaller you make b-a, the higher the likelihood of the sample.
These two properties counterbalance each other. You want to make the support of the function as small as possible without excluding any values. Hence, the MLEs in the one dimensional case are: $\hat a = \min{x_i}, \hat b = \max{x_i}$
Now, for your actual problem, you can take advantage of the independence of the x and y coordinates of each star to break down your 2-D problem into two 1-D problems. Since the window is rectangular, not necessarily a square, the two directions are completely independent and you can simply multiply the likelihoods for the x and y directions to get a joint likelihood of all your parameters.
Hopefully the above gives you some ideas on how to proceed. Comment if anything confuses you.
