Maxwell's theorem
Encyclopedia
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.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, Maxwell's theorem, named in honor of James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
, states that if the probability distribution of a vector
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
-valued random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
X = ( X1, ..., Xn )T is the same as the distribution of GX for every n×n orthogonal matrix
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
G and the components are independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
, then the components X1, ..., Xn are normally distributed with expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
0, all have the same variance, and all are independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
. This theorem is one of many characterizations
Characterization (mathematics)
In mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...
of the normal distribution.
Since a multiplication by an orthogonal matrix is a rotation, the theorem says that if the probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
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
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
0 and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
σ2In, (where In = the n×n identity matrix), for some positive number σ2.