Calculating the Expected Area of a Target Square within a Unit Square

[purplebird]

Homework Statement

Given a unit square and a 'target' square of size LXL = p^2 < 1 in the unit square. The center of target square in equally likely to be anywhere in the unit square. What is the average size of the target square as a function of p^2.

This is the problem and I have included a jpeg illustration of the problem. Any help would be greatly appreciated.

My attempt at the problem :

I divided the problem into three cases : one where the target square is completely in the unit square : one where half the target square is in the unit square and one where quarter of the target square is in the unit square. I then calculated the expectation integral of p^2 but I am confused as to what the probability density function is and what the limits of integral are for each of the cases.

 

[lippyka]

I have tried finding out the probability density functions and limits of integral online and also by myself but I am not able to find them. Any help in this regard would be appreciated.