# 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:

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

$f(x)\geq 0$ • 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

$\int _{a}^{b}f(x)\,dx=P(a\leq x\leq b)$ ## Joint Probability Density Functions

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

{\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

{\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, $f(y)$ . 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 Function

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

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