Cumulative distribution function

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Homework Statement

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

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

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

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

Homework Equations

The Attempt at a Solution

Not sure how to start with A).

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

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

F(x,y) = \int^0_{- \infty} \int^y_{- \infty} f(x,y) dy dx ?

 

[cse63146]

So considering the 3 cases, would it be the following series of integrals:

\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