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:

  • The sum of PMF over all values of x is one:

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.

  • The total area under the PDF is one

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

Joint Probability Density Functions[edit]

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

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

The function

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, . The continuous marginal probability distribution functions are:

and the discrete marginal probability distribution functions are

Conditional Probability Density Function[edit]


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.