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Math Help - A question on joint probability

  1. #1
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    A question on joint probability

    Hi guys, I have a question about joint probability.

    The problem is:
    The joint probability fxn for the continuous random variables X and Y is given by
    f(x,y) = c, 0<=x<=y<=1 (c is a constant),
    = 0 otherwise

    Find the probability that X + Y < 1.

    I was thinking that I'm supposed to take the double integral, from 0 to 1/2 and from x to 1-x, of (c)dydx, since x < y, but I'm not too sure.

    Any help would be greatly appreciated!
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  2. #2
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    Quote Originally Posted by probabilitystudent View Post
    Hi guys, I have a question about joint probability.

    The problem is:
    The joint probability fxn for the continuous random variables X and Y is given by
    f(x,y) = c, 0<=x<=y<=1 (c is a constant),
    = 0 otherwise

    Find the probability that X + Y < 1.

    I was thinking that I'm supposed to take the double integral, from 0 to 1/2 and from x to 1-x, of (c)dydx, since x < y, but I'm not too sure.

    Any help would be greatly appreciated!
    Since the joint pdf is a constant, the probability calculation is simple geometry.

    Shade the region of the xy-plane defined the support defined by 0 \leq x \leq y \leq 1. The area of this region is obviously 1/2.

    Now shade the area of this region enclosed by the y-axis and the line x + y = 1/2. Calculate the area A of this region. Then the required probability is clearly \frac{A}{\frac{1}{2}} = 2A = .....
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  3. #3
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    Sorry, it all makes sense now, but I'm a bit confused about one part. Why is it the line from x+y = 1/2, and not the line x+y < 1
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  4. #4
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    Quote Originally Posted by probabilitystudent View Post
    Sorry, it all makes sense now, but I'm a bit confused about one part. Why is it the line from x+y = 1/2, and not the line x+y < 1
    Misreading on my part. X + Y < 1 it is.
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  5. #5
    MHF Contributor matheagle's Avatar
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    It's just one-half of the region 0<x<y<1

    or \int_0^{.5}\int_x^{1-x}2dydx
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