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Math Help - Double integration: limits

  1. #1
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    Double integration: limits

    I am given the following probability density function:
    f (x,y) = \frac{2x + 2 - y}{4} for 0<x<1 and 0<y<2
    f (x,y) = 0 otherwise

    I am to find P[X + Y > 1].

    My attempts at the double integration failed.

    The answer begins:
    \int_0^1 \int_{1-x}^2 \frac{2x + 2 - y}{4} dydx

    How did they pick which integral was inner?

    How did they choose 1-x as the lower limit on the inner integral?
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  2. #2
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    Here is a graph of the region

    Double integration: limits-capture.jpg

    They solved x+y=1 for y to get y=1-x
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  3. #3
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    Quote Originally Posted by Boris B View Post
    I am given the following probability density function:
    f (x,y) = \frac{2x + 2 - y}{4} for 0<x<1 and 0<y<2
    f (x,y) = 0 otherwise

    I am to find P[X + Y > 1].

    My attempts at the double integration failed.

    The answer begins:
    \int_0^1 \int_{1-x}^2 \frac{2x + 2 - y}{4} dydx

    How did they pick which integral was inner?

    How did they choose 1-x as the lower limit on the inner integral?
    Draw the line x + y = 1 inside the rectangle defined by 0 < x < 1 and 0 < y < 2. This line divides the rectangle into two regions: The region above the line (upper region) and the region below the line (lower region). Note that in the upper region x + y > 1. So you need to set up a double integral of f(x, y) over the upper region .......
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