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Mar22-09, 07:50 PM
P: 452
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] ?
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