Suppose there is a square town, with sides $4$ miles long. A crime is committed with uniform distribution in the town. The police begin at the point $(0,0)$ and go to the point $(x,y)$ where the crime is committed. The roads are rectangular in nature, so the travel distance is $|x|+|y|$. What is the expected travel distance of the police if a crime occurs?
How do I find the desired expectation? I am not sure if $x,y$ are independent and how to treat this.
HINT
Sounds like $(X,Y)$ is distributed uniformly over your square and you would like to calculate $\mathbb{E}[|X|+|Y|]$. Can you isolate the distributions of $X$ and $Y$? Are they independent?
UPDATE
You are correct, since the common distribution is uniform, they can be "untangled" and indeed both $X,Y$ are independent uniform distributions. The question is, the point $(0,0)$ is in the corner of the square or in the center of the square?
If in the corner, then both $X,Y \sim \mathcal{U}(0,4)$ and $X = |X|, Y=|Y|$ which makes your answer correct and the problem quite boring.
On the other hand, I have a suspicion the police are starting in the middle, i.e. $X,Y \sim \mathcal{U}(-2,2)$. Now, $\mathbb{E}[|X|]$ is not so simple to compute. Can you finish?