View source for 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:

* <math> f(x) =\begin{cases}
 P(X = x_i) & \mbox{for } i = 1, 2, \cdots, n \\
 0 & \mbox{for } x \ne x_i
\end{cases}
</math>

*The sum of PMF over all values of ''x'' is one:
:<math>
\sum_i f(x_i)= 1.
</math>

== 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. 
<math>
f(x) \ge 0
</math>
*The total area under the PDF is one
<math>
\int_{- \infty}^{\infty} f(x) \, dx = 1
</math>
*The area under the interval [a, b] is the total probability within this range 
<math>
\int_{a}^{b} f(x) \, dx = P(a \le x \le b)
</math>

== Joint Probability Density Functions ==
Joint '''pdf'''s are ones that are functions of two or more random variables. The function

<math>
\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}
</math>

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

The function

<math>
\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}
</math>

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, <math> f(y) </math>. The continuous marginal probability distribution functions are:

<math>
f(x) = \int_{y}^B f(x,y) dy
</math>

<math>
f(y) = \int_{x}^A f(x,y) dx
</math>

and the discrete marginal probability distribution functions are

<math>
p(x) = \sum_{y \in B} p(x, y)
</math>

<math>
p(y) = \sum_{x \in A} p(x, y)
</math>

== Conditional Probability Density Function ==

<math>
f(x \mid y) = P(X = x, Y = y) = \frac{f(x,y)}{f(y)}
</math>

<math>
f(y \mid x) = P(Y = y, X = x) = \frac{f(x,y)}{f(x)}
</math>

== Statistical Independence ==

*{{cite book
 |last = Gujarati
 |first = D.N.
 |authorlink = 
 |year = 2003
 |title = Basic Econometrics, International Edition - 4th ed.
 |others = 
 |publisher = McGraw-Hill Higher Education 
 |isbn = 0-07-112342-3
 |pages = 870–877
}}

{{BookCat}}