Find the Probability: P(X<1/2 | Y=1)

  1. 1. The problem statement, all variables and given/known data
    Suppose X and Y are jointly continuous random variables with joint density function
    f(x,y)=6x2y, 0<x<y, x+y<2
    f(x,y)=0, otherwise
    Find P(X<1/2 | Y=1).

    2. Relevant equations
    3. The attempt at a solution
    By definition,
    P(X<1/2 | Y=1)
    =∫ fX|Y(x|y=1) dx

    My computations:
    Marginal density of Y:
    fY(y)=2y^4, 0<y<1
    fY(y)=2y(2-y)^3, 1<y<2

    Condition density of X given Y=y:
    Case 1: For given/fixed 0<y<1,
    fX|Y(x|y)=3x^2 / y^3, 0<x<y

    Case 2: For given/fixed 1<y<2,
    fX|Y(x|y)=3x^2 / (2-y)^3, 0<x<2-y

    I hope these are correct. Now P(X<1/2 | Y=1) is the troublesome case because we are given Y=1, which formula for fX|Y(x|y) should I use?

    Thanks for any help!
  2. jcsd
  3. Avodyne

    Avodyne 1,270
    Science Advisor

    Your two formulas are the same at y=1, so it doesn't matter which one you use!
  4. OK, but in general will they always be the same? What should we do in such a case in general?
  5. Avodyne

    Avodyne 1,270
    Science Advisor

    If correctly derived from a given joint density function, yes, they must be the same.
  6. um...Any proof about it?
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