Maxwell's theorem

Maxwell's theorem

In probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a vector-valued random variable X = ( X1, ..., Xn )T is the same as the distribution of GX for every n×n orthogonal matrix G and the components are independent, then the components X1, ..., Xn are normally distributed with expected value 0, all have the same variance, and all are independent. This theorem is one of many characterizations of the normal distribution.

Since a multiplication by an orthogonal matrix is a rotation, the theorem says that if the probability distribution of a random vector is unchanged by rotations and the components are independent, then the components are identically distributed and normally distributed. In other words, the only rotationally invariant probability distributions on Rn that have independent components are multivariate normal distributions with expected value 0 and variance σ2In, (where In = the n×n identity matrix), for some positive number σ2.

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Maxwell–Boltzmann distribution — Maxwell–Boltzmann Probability density function Cumulative distribution function parameters …   Wikipedia

  • Maxwell-Boltzmann-Verteilung — Parameter Definitionsbereich Wahrscheinlichkeitsdichte …   Deutsch Wikipedia

  • Maxwell Rosenlicht — Born April 15, 1924(1924 04 15) Brooklyn Died January 22, 1999(1999 01 22) (aged 74) …   Wikipedia

  • Maxwell Rosenlicht — Maxwell Alexander Rosenlicht (* 15. April 1924 in Brooklyn; † 22. Januar 1999 in Hawaii) war ein US amerikanischer Mathematiker, der sich mit algebraischer Geometrie, algebraischen Gruppen und Differentialalgebra beschäftigte. Maxwell Rosenlicht… …   Deutsch Wikipedia

  • theorem — Maxwell s reciprocity theorem …   Mechanics glossary

  • Maxwell's equations — For thermodynamic relations, see Maxwell relations. Electromagnetism …   Wikipedia

  • Maxwell relations — Thermodynamics …   Wikipedia

  • Maxwell–Boltzmann statistics — Statistical mechanics Thermodynamics · …   Wikipedia

  • Maxwell's equations in curved spacetime — Induced spacetime curvature In physics, Maxwell s equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not… …   Wikipedia

  • Maxwell speed distribution — Classically, an ideal gas molecules bounce around with somewhat arbitrary velocities, never interacting with each other. In reality, however, an ideal gas is subjected to intermolecular forces. It is to be noted that the aforementioned classical… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”