probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, Maxwell's theorem, named in honor of

James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...

, states that if the probability distribution of a

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

-valued

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

''X'' = ( ''X''₁, ..., ''X''_''n'' )^''T'' is the same as the distribution of ''GX'' for every ''n''×''n''

orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ...

''G'' and the components are

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...

, then the components ''X''₁, ..., ''X''_''n'' are

normally distributed In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is ...

with

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

0 and all have the same

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

. 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 In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

of a random vector is unchanged by rotations and if the components are independent, then the components are identically distributed and normally distributed. In other words, the only rotationally invariant probability distributions on R^''n'' that have independent components are

multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...

s with

0 and

''σ''²''I''_''n'', (where ''I''_''n'' = the ''n''×''n'' identity matrix), for some positive number ''σ''².

References

* * {{DEFAULTSORT:Maxwell's Theorem Probability theorems James Clerk Maxwell