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Multiple Random Variable Question

  1. Nov 10, 2013 #1
    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 [itex]Z[/itex] BUT, since there are several approaches here goes the method of choice:

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

    now I just need [itex]F_Z(z)[/itex] and [itex]-1 <= z <= 1[/itex]

    I know I need to split up when [itex]-1 <= z <= 0[/itex] and [itex]0 < z <= 1[/itex]

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

    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 : [itex]F_Z(z) = \int_{1}^{2-z} \int_{{y+z}}^2 1 \,dy\,dx = 1/2(1+z)^2[/itex]

    I am picturing a square over the coordinates [itex](1,1),(1,2),(2,1),(2,2)[/itex] and [itex]Z = X - Y[/itex] being a line going through it and intersecting [itex](1,1),(2,2)[/itex]. 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
  2. jcsd
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