what is the difference between the joint probability density function and the joint distribution function when they are both denoted as F(x,y)?

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- Mar 21st 2010, 04:22 AMalexandrabel90definitions
what is the difference between the joint probability density function and the joint distribution function when they are both denoted as F(x,y)?

- Mar 21st 2010, 11:02 AMCaptainBlack
The joint density is normally a lower case function like $\displaystyle f(x,y)$, then:

$\displaystyle P((x,y) \in R)=\int_R f(x,y)\; dxdy$

The joint distribution function is usually an upper case function like $\displaystyle F(x,y)$, and is defined as:

$\displaystyle F(x,y)=P(X<x, Y<y)=\int_{\xi=-\infty}^y \int_{\zeta=-\infty}^x f(\zeta,\xi)\; d\zeta d\xi$

CB