Joint Density

Consider two random variables \(X\) and \(Y\). The probability of some event \(A\) that lies in the domain of both \(X\) and \(Y\) can be expressed as,

\[ P((X,Y) \in A) = \iint_A f(x,y)~ dxdy \]

Where \(f(x,y)\) is the joint probability density function

Independence

If \(X\) and \(Y\) are independent then their joint PDF is separable,

\[ f(x,y) = f_X(x)f_Y(y) \]

Marginal Density

The PDF of \(X\) can be solved using the marginal density.

\[ f_X(x) = \int_y f(x,y)~ dy \]

Conditional Density

The conditoinal density can be determined using the divison rule,

\[ f(y \mid x) = \frac{f(x,y)}{f_X(x)} \]