# Cumulative distribution function

#### cse63146

1. 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 $$F_Y (y)$$
2. Homework Equations

3. The Attempt at a Solution

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:

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

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#### cse63146

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

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

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