The random variable XX is said to be a Gaussian random variable 1 if its probability density function has the form

The mean of such a Gaussian random variable is mm and its variance σ2 σ 2 . As a shorthand notation, this information is denoted by. x∼𝒩mσ2 x m σ 2 . The characteristic function Φ X ⁢· Φ X · of a Gaussian random variable is given by Φ X ⁢i⁢u=ei⁢m⁢u⁢e−σ2⁢u22 Φ X u m u σ 2 u 2 2

No closed form expression exists for the probability distribution function of a Gaussian random variable. For a zero-mean, unit-variance, Gaussian random variable 𝒩01 0 1 , the probability that it exceeds the value xx is denoted by Q⁢x Q x . PrX>x=1−PXx=12⁢π⁢∫x∞e−α22dα≡Q⁢x X x 1 P X x 1 2 α x α 2 2 Q x

A plot of Q⁢· Q · is shown in Figure 1. When the Gaussian random variable has non-zero mean and/or non-unit variance, the probability of it exceeding xx can also be expressed in terms of Q⁢· Q · . Integrating by parts, Q⁢· Q · is bounded (for x>0 x 0 ) by As xx becomes large, these bounds approach each other and either can serve as an approximation to Q⁢· Q · ; the upper bound is usually chosen because of its relative simplicity. The lower bound can be improved; noting that the term x1+x2 x 1 x 2 decreases for x<1 x 1 and that Q⁢x Q x increases as xx decreases, the term can be replaced by its value at x=1 x 1 without affecting the sense of the bound for x≤1 x 1 .

We will have occasion to evaluate the expected value of ea⁢X+b⁢X2 a X b X 2 where X∼𝒩mσ2 X m σ 2 and aa, bb are constants. By definition, Eea⁢X+b⁢X2=12⁢π⁢σ2⁢∫−∞∞ea⁢x+b⁢x2−x−m22⁢σ2dx a X b X 2 1 2 σ 2 x a x b x 2 x m 2 2 σ 2 The argument of the exponential requires manipulation (i.e., completing the square) before the integral can be evaluated. This expression can be written as −(12⁢σ2⁢((1−2⁢b⁢σ2)⁢x2−2⁢(m+a⁢σ2)⁢x+m2)) 1 2 σ 2 1 2 b σ 2 x 2 2 m a σ 2 x m 2 Completing the square, this expression can be written −(1−2⁢b⁢σ22⁢σ2⁢x−m+a⁢σ21−2⁢b⁢σ22)+1−2⁢b⁢σ22⁢σ2⁢m+a⁢σ21−2⁢b⁢σ22−m22⁢σ2 1 2 b σ 2 2 σ 2 x m a σ 2 1 2 b σ 2 2 1 2 b σ 2 2 σ 2 m a σ 2 1 2 b σ 2 2 m 2 2 σ 2 We are now ready to evaluate the integral. Using this expression, Eea⁢X+b⁢X2=e1−2⁢b⁢σ22⁢σ2⁢m+a⁢σ21−2⁢b⁢σ22−m22⁢σ2⁢12⁢π⁢σ2⁢∫−∞∞e−(1−2⁢b⁢α22⁢σ2⁢x−m+a⁢σ21−2⁢b⁢σ22)dx a X b X 2 1 2 b σ 2 2 σ 2 m a σ 2 1 2 b σ 2 2 m 2 2 σ 2 1 2 σ 2 x 1 2 b α 2 2 σ 2 x m a σ 2 1 2 b σ 2 2 Let α=x−m+a⁢σ21−2⁢b⁢σ2σ1−2⁢b⁢σ2 α x m a σ 2 1 2 b σ 2 σ 1 2 b σ 2 which implies that we must require that 1−2⁢b⁢σ2>0 1 2 b σ 2 0 (or b<12⁢σ2 b 1 2 σ 2 ). We then obtain Eea⁢X+b⁢X2=e1−2⁢b⁢σ22⁢σ2⁢m+a⁢σ21−2⁢b⁢σ22−m22⁢σ2⁢11−2⁢b⁢σ2⁢12⁢π⁢∫−∞∞e−α22dα a X b X 2 1 2 b σ 2 2 σ 2 m a σ 2 1 2 b σ 2 2 m 2 2 σ 2 1 1 2 b σ 2 1 2 α α 2 2 The integral equals unity, leaving the result

Important special cases are

The real-valued random vector X X is said to be a Gaussian random vector if its joint distribution function has the form

If complex-valued, the joint distribution of a circular Gaussian random vector is given by The vector m X m X denotes the expected value of the Gaussian random vector and KX KX its covariance matrix. m X =EX m X X K X =EX⁢XT− m X ⁢ m X T K X X X m X m X As in the univariate case, the Gaussian distribution of a random vector is denoted by X∼𝒩 m X K X X m X K X . After applying a linear transformation to Gaussian random vector, such as Y=A⁢X Y A X , the result is also a Gaussian random vector (a random variable if the matrix is a row vector): Y∼𝒩A⁢ m X A⁢ K X ⁢AT Y A m X A K X A . The characteristic function of a Gaussian random vector is given by Φ X ⁢i⁢u=ei⁢uT⁢ m X −12⁢uT⁢ K X ⁢u Φ X u u m X 1 2 u K X u From this formula, the N th N th -order moment formula for jointly distributed Gaussian random variables is easily derived. 2 E X 1 … X N ={∑ 𝒫 N 𝒫 N E X 𝒫 N ( 1 ) ⁢ X 𝒫 N ( 2 ) ⁢…⁢E X 𝒫 N ( N - 1 ) ⁢ X 𝒫 N ( N )   if  N   even∑ 𝒫 N 𝒫 N E X 𝒫 N ( 1 ) ⁢E X 𝒫 N ( 2 ) ⁢ X 𝒫 N ( 3 ) ⁢…⁢E X 𝒫 N ( N - 1 ) ⁢ X 𝒫 N ( N )   if  N   odd X 1 … X N 𝒫 N 𝒫 N X 𝒫 N ( 1 ) X 𝒫 N ( 2 ) … X 𝒫 N ( N - 1 ) X 𝒫 N ( N ) N   even 𝒫 N 𝒫 N X 𝒫 N ( 1 ) X 𝒫 N ( 2 ) X 𝒫 N ( 3 ) … X 𝒫 N ( N - 1 ) X 𝒫 N ( N ) N   odd where 𝒫 N 𝒫 N denotes a permutation of the first NN integers and 𝒫 N ⁢i 𝒫 N i the i th i th element of the permutation. For example, E X 1 ⁢ X 2 ⁢ X 3 ⁢ X 4 =E X 1 ⁢ X 2 ⁢E X 3 ⁢ X 4 +E X 1 ⁢ X 3 ⁢E X 2 ⁢ X 4 +E X 1 ⁢ X 4 ⁢E X 2 ⁢ X 3 X 1 X 2 X 3 X 4 X 1 X 2 X 3 X 4 X 1 X 3 X 2 X 4 X 1 X 4 X 2 X 3 .

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This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Charlet Reedstrom on Aug 4, 2003 4:20 pm -0500.