## Probability Mass Function of a Discrete Random VariableEdit

A **probability mass function** *f(x)* (PMF) of *X* is a function that determines the probability in terms of the input variable *x*, which is a discrete **random variable** (rv).

A pmf has to satisfy the following properties:

- The sum of PMF over all values of
*x*is one:

## Probability Density Function of a Continuous Random VariableEdit

The continuous PDF requires that the input variable *x* is now a continuous rv. The following conditions must be satisfied:

- All values are greater than zero.

- The total area under the PDF is one

- The area under the interval [a, b] is the total probability within this range

## Joint Probability Density FunctionsEdit

Joint **pdf**s are ones that are functions of two or more random variables. The function

is the **continuous joint probability density function**. It gives the joint probability for x and y.

The function

is similarly the **discrete joint probability density function**

## Marginal Probability Density FunctionEdit

The marginal **PDFs** are derived from the joint PDFs. If the joint pdf is integrated over the distribution of the X variable, then one obtains the marginal PDF of y, . The continuous marginal probability distribution functions are:

and the discrete marginal probability distribution functions are

## Conditional Probability Density FunctionEdit

## Statistical IndependenceEdit

- Gujarati, D.N. (2003).
*Basic Econometrics, International Edition - 4th ed.*. McGraw-Hill Higher Education. pp. 870-877. ISBN 0-07-112342-3.