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Math Help - Joint pdf Problem??

  1. #1
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    Joint pdf Problem??

    Let T1 and T2 be random times for a company to complete two steps in a certain process. Say T1 and T2 are measured in days and they have the joint p.d.f. that is uniform over the space 1 < t1 < 10, 2 < t2 < 6, t1 + 2t2 < 14. What is the P( T1 + T2 > 10)?
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  2. #2
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    Quote Originally Posted by jojo_jojo View Post
    Let T1 and T2 be random times for a company to complete two steps in a certain process. Say T1 and T2 are measured in days and they have the joint p.d.f. that is uniform over the space 1 < t1 < 10, 2 < t2 < 6, t1 + 2t2 < 14. What is the P( T1 + T2 > 10)?
    Thanks
    The line T1+ 2T2= 14 is a diagonal of the rectangle with vertices (1,2), (10, 4), (10,6), and (1,6). The joint p.d.f is a constant, 1 over area of the triangle with vertices (1,2), (10,4), and (1,6).

    Graph that region and the line T1+ T2= 10. That will cut the region above in two points, where T1+ T2= 10 intersects T1+ 2T2= 15, and where T1+ T2= 10 intersects T2= 2. P(T1+T2> 10) is the constant above multiplied by the area of the triangle with those two points and (10, 2) as vertices.
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  3. #3
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    Thanks for the quick reply, but would you please explain your answer a little bit. I mean I understand how you got the 3 vertices, but couldn't understand how you got (10,4). Second, I also unable to understand last 2 lines.
    Last edited by jojo_jojo; March 13th 2009 at 01:57 PM.
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  4. #4
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    I misread your post! That "4" shoul be 6.
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  5. #5
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    Ya now it makes sense, would you please explain your this statement (I think here 15 is typo) also:

    Graph that region and the line T1+ T2= 10. That will cut the region above in two points, where T1+ T2= 10 intersects T1+ 2T2= 15, and where T1+ T2= 10 intersects T2= 2. P(T1+T2> 10) is the constant above multiplied by the area of the triangle with those two points and (10, 2) as vertices.

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  6. #6
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    Did you draw a picture? This is why you studied geometry and the calculus of multiple variables. Don't let the opportunity get away!

    The area of the entire region:

    \int_{1}^{2}6-2\;d(t1) + \int_{2}^{10}(7-\frac{t1}{2}) - 2\;d(t1)

    This gives your uniform depth.

    Now find the volume of that other piece.

    Of course, since it's Uniform, it really masquerades as a Single Variable problem. Anything else and it's a different world.
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