# Econometric Theory/Probability Density Function (PDF)

## Probability Mass Function of a Discrete Random Variable

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:

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

## Probability Density Function of a Continuous Random Variable

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.

${\displaystyle f(x)\geq 0}$

• The total area under the PDF is one

${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=1}$

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

${\displaystyle \int _{a}^{b}f(x)\,dx=P(a\leq x\leq b)}$

## Joint Probability Density Functions

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

{\displaystyle {\begin{aligned}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{aligned}}}

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

The function

{\displaystyle {\begin{aligned}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{aligned}}}

is similarly the discrete joint probability density function

## Marginal Probability Density Function

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, ${\displaystyle f(y)}$. The continuous marginal probability distribution functions are:

${\displaystyle f(x)=\int _{y}^{B}f(x,y)dy}$

${\displaystyle f(y)=\int _{x}^{A}f(x,y)dx}$

and the discrete marginal probability distribution functions are

${\displaystyle p(x)=\sum _{y\in B}p(x,y)}$

${\displaystyle p(y)=\sum _{x\in A}p(x,y)}$

## Conditional Probability Density Function

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

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

## Statistical Independence

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