• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Cumulative distribution function

  • Thread starter cse63146
  • Start date
1. Homework Statement

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. Homework 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] ?
 
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]
 

Related Threads for: Cumulative distribution function

  • Posted
Replies
3
Views
1K
  • Posted
Replies
1
Views
6K
Replies
9
Views
634
Replies
6
Views
5K
Replies
2
Views
862

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top