# Multiple Random Variable Question

1. Nov 10, 2013

### mynameisfunk

1. The problem statement, all variables and given/known data
A and B agree to meet at a certain place between 1 PM and 2 PM. Suppose they arrive at the meeting place independently and randomly during the hour. find the distribution of the length of time that A waits for B. (If B arrives before A, define A's waiting time as 0.)

2. Relevant equations

3. The attempt at a solution

My professor provides the solutions to my homework questions, therefore, I have it but I do NOT understand. I'm working on finding the CDF of $Z$ BUT, since there are several approaches here goes the method of choice:

$X =$time of arrival for B
$Y =$time of arrival for A
$Z = X - Y$ defines the waiting time for A with $-1 <= x <= 1$ and$-1 <= y <= 1$

now I just need $F_Z(z)$ and $-1 <= z <= 1$

I know I need to split up when $-1 <= z <= 0$ and $0 < z <= 1$

the first part
$F_Z(z) = \int_{1}^{2+z} \int_{{x-z}}^2 1 \,dy\,dx = (1/2)*(1+z)^2$

However, the second part, my professor did something weird. My latex skills are horrible, so I just posted the picture.

Now, I cannot understand why we wouldn't just do this : $F_Z(z) = \int_{1}^{2-z} \int_{{y+z}}^2 1 \,dy\,dx = 1/2(1+z)^2$

I am picturing a square over the coordinates $(1,1),(1,2),(2,1),(2,2)$ and $Z = X - Y$ being a line going through it and intersecting $(1,1),(2,2)$. The upper part is where A doesn't wait and the lower half is where A does wait.

I'm sorry for my horrible question asking skills.. Let me know if I need to provide more info! Thanks so much guys..

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Last edited: Nov 10, 2013