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Cumulative distribution function

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Let X and Y be continuous random variables having joint probability density function

    [tex]f(x,y) = e^{-y}[/tex] if 0 [tex]\leq x \leq y[/tex]

    A) Determine the joint cumulative distribution function F(x,y) of X and Y. (Hint: Consider the three cases 1) [tex]x \leq 0[/tex] or [tex]y \leq 0[/tex] 2) 0 < x < y 3) 0 <y < x

    B) Let [tex]F_X (x)[/tex] and [tex]F_Y (y)[/tex] be the marginal cumulative distribution functions of X and Y. One can show that [tex]F_X (x) = Lim_{Y \rightarrow \infty} F(x,y)[/tex] and [tex]F_Y (y) = Lim_{X \rightarrow \infty} F(x,y)[/tex]. Use this result to obtain [tex]F_X (x)[/tex] and [tex]F_Y (y)[/tex]
    2. Relevant equations



    3. The attempt at a solution

    Not sure how to start with A).

    I know that [tex] F(x,y) = \int^x_{- \infty} \int^y_{- \infty} f(x,y) dy dx[/tex]

    Does it mean for the case where x < 0 it would be:

    [tex] F(x,y) = \int^0_{- \infty} \int^y_{- \infty} f(x,y) dy dx[/tex] ?
     
  2. jcsd
  3. Mar 23, 2009 #2
    So considering the 3 cases, would it be the following series of integrals:

    [tex]\int^x_{- \infty}\int^0_{- \infty}f(u,v) dv du + \int^x_{- \infty}\int^y_{x}f(u,v) dv du + \int^x_{- \infty}\int^x_{y}f(u,v) dv du[/tex]
     
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