Get Expectation Of Quadratic Forms In Normal And Nonnormal - Emlab Berkeley

q(x) = A1q1(x) + ··· + Anqn(x) for some set of coefficients, we say that this expression is a normal form of q. (k ∈ K, X ∈ V ). If D denotes the space of diagonal matrices in V , then the principal axes theorem asserts that V = K · D := {k · d | k ∈ K, d ∈ D}.

The quadratic form can be written as where we have defined By the above theorem on orthogonal transformations of standard multivariate normal random vectors, the orthogonality of implies that . Since is diagonal, we can write the quadratic form as where is the -th component of and is the -th diagonal entry of .

4.3 Multivariate Normal. (4.17) Q = ( x − μ ) T V − 1 ( x − μ ) , is called the quadratic form of the multivariate normal distribution. The distribution possesses a number of important properties, and three are discussed below.

In general, there will be many different matrices corresponding to the same quadratic form. However, if we add the condition that a matrix be symmetric, then we do have uniqueness.

Let A be such a matrix, and let X ∼ Nn(��, ��) with �� > 0. The scalar random vari- able (hereafter abbreviated r.v.) Y = X′AX is referred to as a quadratic form (in normal variables).

q(x) = A1q1(x) + ··· + Anqn(x) for some set of coefficients, we say that this expression is a normal form of q. (k ∈ K, X ∈ V ). If D denotes the space of diagonal matrices in V , then the principal axes theorem asserts that V = K · D := {k · d | k ∈ K, d ∈ D}.

The quadratic form is a special nonlinear function having only second-order terms (either the square of a variable or the product of two variables). From: Introduction to Optimum Design (Third Edition), 2012.