How Do You Determine the Joint CDF for Joy and Ethan's Arrival Times?

[marcadams267]

So I have this word problem as seen below:

Joy and Ethan have agreed to meet for dinner between 8:00 PM and 9:00 PM. Suppose that Ethan may
arrive at any time between the set meeting. Joy on the other hand will arrive at the set meeting under the
following conditions:
• Joy will always arrive earlier than Ethan.
• Joy will never arrive later than 20 minutes.
• Joy’s arrival time added to Ethan’s arrival time will never exceed an hour.
Let X be the arrival time (in minutes) of Ethan and Y be the arrival time (in minutes) of Joy after 8:00
PM. Assume that all possible arrival times under the specified conditions are equally likely to occur.

I need to find the joint PDF of X and Y, which is easy enough as all that requires is differentiating the joint CDF with respect to both x and y. However, I'm not entirely sure how I would go about finding the joint CDF from the word problem above.
So far, I've gotten the bounds of the two variables as 0<X< 20 and X<Y<60 and X+Y <=60.
However, I do not know what to do next. I would appreciate any help on how to go about solving this problem. Thank you.

 

[Valkarie]

A:The joint CDF is$$F_{X,Y}(x,y) = P\{X \leq x, Y \leq y\}$$and the PDF is$$f_{X,Y}(x,y) = \frac{\partial^2 F_{X,Y}}{\partial x \partial y}.$$For $y>x$, we have\begin{align}F_{X,Y}(x,y) &= \int_0^x\int_t^y f_{X,Y}(s,t)\;ds\;dt\\&= \int_0^x\int_t^y f_X(s) f_Y(t|s)\;ds\;dt\\&= \int_0^x f_X(s)\left[\int_t^y f_Y(t|s)\;dt\right]\;ds\\&= \int_0^x f_X(s)(1-F_Y(t|s))\;ds.\end{align}Where$$f_X(x) = \frac{1}{20},\quad 0 < x < 20,$$is the PDF of $X$ and$$F_Y(y|x) = \begin{cases}0, & 0 \leq y \leq x,\\\frac{y-x}{60-x}, & x < y \leq 60,\end{cases}$$is the conditional CDF of $Y$ given $X$. The PDF of $Y$ conditioned on $X$ is$$f_Y(y|x) = \frac{1}{60-x},\quad x < y \leq 60.$$So\begin{align}F_{X,Y}(x,y) &= \int_0^x \frac{1}{20}\left(1-\frac{y-s}{60-s}\right)\;ds\\&= \frac{1}{20}\left[x - \frac{y-x}{60-x}\right].\end{align}Differentiating twice with respect to $x$ and