Wishart Distribution - Probability Density Function

Probability Density Function

The Wishart distribution can be characterized by its probability density function, as follows.

Let be a p × p symmetric matrix of random variables that is positive definite. Let V be a (fixed) positive definite matrix of size p × p.

Then, if n ≥ p, has a Wishart distribution with n degrees of freedom if it has a probability density function given by

where Γp(·) is the multivariate gamma function defined as

\Gamma_p(n/2)= \pi^{p(p-1)/4}\Pi_{j=1}^p \Gamma\left.

In fact the above definition can be extended to any real n > p − 1. If n ≤ p − 2, then the Wishart no longer has a density—instead it represents a singular distribution.

Read more about this topic:  Wishart Distribution

Famous quotes containing the words probability and/or function:

    “Liberty is a blessing so inestimable, that, wherever there appears any probability of recovering it, a nation may willingly run many hazards, and ought not even to repine at the greatest effusion of blood or dissipation of treasure.”
    —David Hume (1711–1776)

    “Uses are always much broader than functions, and usually far less contentious. The word function carries overtones of purpose and propriety, of concern with why something was developed rather than with how it has actually been found useful. The function of automobiles is to transport people and objects, but they are used for a variety of other purposes—as homes, offices, bedrooms, henhouses, jetties, breakwaters, even offensive weapons.”
    —Frank Smith (b. 1928)