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Math Help - find the probability that Y1 + Y2 is less than 1

  1. #1
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    find the probability that Y1 + Y2 is less than 1

    Suppose that the random variables Y1 and Y2 have a joint probability distribution function f(y1, y2) given by

    f(y1,y2)= 6y1^2y2  0<=y1<=y2, y1+y2<=2 and f(y1,y2)=0 elsewhere

    (A) What is the probability that Y1 + Y2 is less than 1?

    for this question, i have hard time findthe region of my integration, it seems like i need to use separate integral to do this, if we do it as dy2dy1 order, then y2 goes from y1 to 1-y1, but then y2 go from 0 to to separate function, so do i need to do it twice?
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  2. #2
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    Re: find the probability that Y1 + Y2 is less than 1

    Quote Originally Posted by wopashui View Post
    Suppose that the random variables Y1 and Y2 have a joint probability distribution function f(y1, y2) given by

    f(y1,y2)= 6y1^2y2 0<=y1<=y2, y1+y2<=2 and f(y1,y2)=0 elsewhere

    (A) What is the probability that Y1 + Y2 is less than 1?

    for this question, i have hard time findthe region of my integration, it seems like i need to use separate integral to do this, if we do it as dy2dy1 order, then y2 goes from y1 to 1-y1, but then y2 go from 0 to to separate function, so do i need to do it twice?
    Draw the region defined by the support. You will need to use your knowledge of multi-variable calculus, in particular double integrals. Your textbook or class notes should have exmaples to follow.
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    Re: find the probability that Y1 + Y2 is less than 1

    i have drawn the region, but y1 would go to something depends on y2 eventually, it will end up with a function of y2, but we need a number here

    we are given the answer is 1/32, i know y2 goes from 0 to y1 to 1-y1, but y1 will goes from 0 to some 1-y2, and 0 to y2, which will end up with a function of y2
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    Re: find the probability that Y1 + Y2 is less than 1

    Are you SURE you are considering the correct region.

    Checking \int_{0}^{1}\int_{x_{1}}^{2-x_{1}}\;6\cdot x_{1}^{2}\cdot x_{2}\;\;dx_{2}\;dx_{1}\;=\;1

    Excellent. This indicates we might be on the right track.

    Make two modifications to the integral and you are done.

    Show us what you get.
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  5. #5
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    Re: find the probability that Y1 + Y2 is less than 1

    it should be 0 to 0.5 and y1 to 1-y1
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