Calculating Average Size of Target Square in Unit Square with Variable p^2

[purplebird]

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.

 

[Pere Callahan]

Given a unit square and a 'target' square of size LXL = p^2 < 1 in the unit square. ... What is the average size of the target square as a function of p^2.

The size of the "target square" is p^2 by the statement of the problem.

Do you mean the intersection of the target square T with the unit square $U=[0,1]\times[0,1]\subset \mathbb{R}^2$? In this case I suggest you first try and find out what the area of $A = T\cap U$ is as a function of the center of the target square. Denote this center by (x,y).

Then for $p \leq x\leq 1-p \wedge p \leq y\leq 1-p$ you have A=p^2. Figure out the area A if the center of the target space is too close to the boundary of the unit square for the former to completely fit into the latter.