Econometric Theory/Probability Density Function (PDF)

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Probability Mass Function of a Discrete Random Variable[edit]

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 Variable[edit]

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 Functions[edit]

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 Function[edit]

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[edit]


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[edit]

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