Multivariate random variable

Multivariate random variable

In mathematics, probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose values is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value.

More formally, a multivariate random variable is a column vector X = (X1, ..., Xn)T (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space (Ω, \scriptstyle \mathcal{F}, P), where Ω is the sample space, \scriptstyle \mathcal{F} is the sigma-algebra (the collection of all events), and P is the probability measure (a function returning every event's probability).

Contents

Probability distribution

Every random vector gives rise to a probability measure on Rn with the Borel algebra as the underlying sigma-algebra. This measure is also known as the joint probability distribution, the joint distribution, or the multivariate distribution of the random vector.

The distributions of each of the component random variables Xi are called marginal distributions. The conditional probability distribution of Xi given Xj is the probability distribution of Xi when Xj is known to be a particular value.

Operations on random vectors

Random vectors can be subjected to the same kinds of algebraic operations as can non-random vectors: addition, subtraction, multiplication by a scalar, and the taking of inner products.

Expected value, covariance, and cross-covariance

The expected value or mean of a random vector X is a fixed vector E(X) whose elements are the expected values of the respective random variables.

The covariance matrix (also called the variance-covariance matrix) of an n× 1 random vector is an n × n matrix whose i, j element is the covariance between the ith and the jth random variables. The covariance matrix is the expected value, element by element, of the n × n matrix computed as [X – E(X)][X-E(X)]T, where the superscript T refers to the transpose of the indicated vector:

\operatorname{Var}(X)=\operatorname{E}{[X-\operatorname{E}(X)][X-\operatorname{E}(X)]^{T}}.

By extension, the cross-covariance matrix between two random vectors X and Y (X having n elements and Y having p elements) is the n × p matrix

\operatorname{Cov}(X,Y)=\operatorname{E}[X-\operatorname{E}(X)][Y-\operatorname{E}(Y)]^{T},

where again the indicated matrix expectation is taken element-by-element in the matrix. The cross-covariance matrix Cov(Y, X) is simply the transpose of the matrix Cov(X, Y).

Further properties

One can take the expectation of a quadratic form in the random vector X as follows:[1]:p.170-171

\operatorname{E}(X^{T}AX) = [\operatorname{E}(X)]^{T}A[\operatorname{E}(X)] + \operatorname{tr}(AC),

where C is the covariance matrix of X and tr refers to the trace of a matrix — that is, to the sum of the elements on its main diagonal (from upper left to lower right). Since the quadratic form is a scalar, so is its expectation.

One can take the expectation of the product of two different quadratic forms in a zero-mean Gaussian random vector X as follows:[1]:pp. 162-176

\operatorname{E}[X^{T}AX][X^{T}BX] = 2\operatorname{tr}(ACBC) + \operatorname{tr}(AC)\operatorname{tr}(BC)

where again C is the covariance matrix of X. Again, since both quadratic forms are scalars and hence their product is a scalar, the expectation of their product is also a scalar.

Applications

Portfolio theory

In portfolio theory in finance, an objective often is to choose a portfolio of risky assets such that the distribution of the random portfolio return has desirable properties. For example, one might want to choose the portfolio return having the lowest variance for a given expected value. Here the random vector is the vector r of random returns on the individual assets, and the portfolio return p (a random scalar) is the inner product of the vector of random returns with a vector w of portfolio weights — the fractions of the portfolio placed in the respective assets. Since p = wTr, the expected value of the portfolio return is wTE(r) and the variance of the portfolio return can be shown to be wTCw, where C is the covariance matrix of r.

Regression theory

In linear regression theory, we have data on n observations on a dependent variable y and n observations on each of k independent variables xj. The observations on the dependent variable are stacked into a column vector y; the observations on each independent variable are also stacked into column vectors, and these latter column vectors are combined into a matrix X of observations on the independent variables. Then the following regression equation is postulated as a description of the process that generated the data:

y = X \beta + e,\,

where \beta\, is a postulated fixed but unknown vector of k response coefficients, and e is an unknown random vector reflecting random influences on the dependent variable. By some chosen technique such as ordinary least squares, a vector \hat \beta is chosen as an estimate of \beta\,, and the estimate of the vector e\,, denoted \hat e, is computed as

\hat e = y - X \hat \beta.

Then the statistician must analyze the properties of \hat \beta and \hat e, which are viewed as random vectors since a randomly different selection of n cases to observe would have resulted in different values for them.

References

  1. ^ a b Kendrick, David, Stochastic Control for Economic Models, McGraw_Hill, 1981.

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Multivariate — may refer to: in Mathematics Multivariable calculus Multivariate division algorithm Multivariate interpolation Multivariate polynomial in Statistics Multivariate analysis Multivariate random variable Multivariate statistics in other areas… …   Wikipedia

  • Multivariate stable distribution — multivariate stable Probability density function Heatmap showing a Multivariate (bivariate) stable distribution with α = 1.1 parameters: exponent shift/location vector …   Wikipedia

  • Multivariate normal distribution — MVN redirects here. For the airport with that IATA code, see Mount Vernon Airport. Probability density function Many samples from a multivariate (bivariate) Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the… …   Wikipedia

  • Random matrix — In probability theory and mathematical physics, a random matrix is a matrix valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a… …   Wikipedia

  • Multivariate analysis of variance — (MANOVA) is a generalized form of univariate analysis of variance (ANOVA). It is used when there are two or more dependent variables. It helps to answer : 1. do changes in the independent variable(s) have significant effects on the dependent …   Wikipedia

  • Multivariate statistics — is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis. Methods of bivariate statistics, for example simple linear… …   Wikipedia

  • Multivariate adaptive regression splines — (MARS) is a form of regression analysis introduced by Jerome Friedman in 1991.[1] It is a non parametric regression technique and can be seen as an extension of linear models that automatically models non linearities and interactions. The term… …   Wikipedia

  • Variable régionalisée — La VR comme phénomène physique : topographie de la ville de Binche …   Wikipédia en Français

  • Multivariate mutual information — In information theory there have been various attempts over the years to extend the definition of mutual information to more than two random variables. These attempts have met with a great deal of confusion and a realization that interactions… …   Wikipedia

  • Multivariate kernel density estimation — Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental questions in statistics. It can be viewed as a generalisation of histogram density… …   Wikipedia

Share the article and excerpts

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