# Econometric Theory/Probability Density Function (PDF)

## 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:

• $f(x) =\begin{cases} P(X = x_i) & \mbox{for } i = 1, 2, \cdots, n \\ 0 & \mbox{for } x \ne x_i \end{cases}$
• The sum of PMF over all values of x is one:
$\sum_i f(x_i)= 1.$

## 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.

$f(x) \ge 0$

• The total area under the PDF is one

$\int_{- \infty}^{\infty} f(x) \, dx = 1$

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

$\int_{a}^{b} f(x) \, dx = P(a \le x \le b)$

## Joint Probability Density FunctionsEdit

Joint pdfs are ones that are functions of two or more random variables. The function

\begin{align} f(X \in A, Y \in B) & = \int_{A} \, \int_{B} f(x,y) \, dx \, dy \\ & = 0, \mbox{if } x \notin A \mbox{ and } y \notin B \\ \end{align}

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

The function

\begin{align} p(X \in A, Y \in B) & = \sum_{X \in A} \sum_{Y \in B} p(x, y) \\ & = 0, \mbox{if } x \notin A \mbox{ and } Y \notin y \\ \end{align}

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, $f(y)$. The continuous marginal probability distribution functions are:

$f(x) = \int_{y}^B f(x,y) dy$

$f(y) = \int_{x}^A f(x,y) dx$

and the discrete marginal probability distribution functions are

$p(x) = \sum_{y \in B} p(x, y)$

$p(y) = \sum_{x \in A} p(x, y)$

## Conditional Probability Density FunctionEdit

$f(x \mid y) = P(X = x, Y = y) = \frac{f(x,y)}{f(y)}$

$f(y \mid x) = P(Y = y, X = x) = \frac{f(x,y)}{f(x)}$

## Statistical IndependenceEdit

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