Two people have decided to meet at a certain point in a forest sometime between noon and 2pm. Their respective independent arrival times are $X$ and $Y$ such that $X \sim \mathrm{Unif}(0,2)$ and $Y \sim \mathrm{Unif}(0,2)$.
Hence the joint density of $X$ and $Y$ is $f_{X,Y} {(x,y)} = \begin{cases} 1/4, & 0< x <2 , 0< y <2 \\ 0, & \text{otherwise.} \end{cases} $
They have agreed that whoever arrives first will wait for at most $20$ minutes for the arrival of the other.
a) Sketch the region in the $xy$ plane of times values for which they will meet and specify precisely the appropriate bounds (in terms of $x$ and $y$) for this region; then find the probabilty that they will meet by integrating the joint PDF $f_{X,Y} {(x,y)}$ over this region.
b) Since $X$ and $Y$ are independent, what value must $\mathrm{Cov}(X,Y)$ have?
c) Calculate explicitly $\mathrm{Cov}(X,Y)$ starting from its definition. Recall that $\mathrm{Cov}(X,Y) = E[(X - E(X))(Y - E(Y))].$
I know this is quite a long question but I didn't know how to break it down into smaller parts without just having to type it into three different questions to ask. If you could give me an idea about the sketch great! I'm not sure how to integrate the PDF as it's only $1/4$? Would it not just be $x/4 + C$?
Also for $\mathrm{Cov}(X,Y)$, I don't seem to have any notes on this, so detail would be good too.
Test is in the morning and you guys have been a big help so far!