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# The Gaussian Random Variable

Module by: Don Johnson. E-mail the author

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

pXx=12πσ2exm22σ2 p X x 1 2 σ 2 x m 2 2 σ 2
(1)
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. x𝒩mσ2 x m σ 2 . The characteristic function Φ X · Φ X · of a Gaussian random variable is given by Φ X iu=eimueσ2u22 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 Qx Q x . PrX>x=1PXx=12πxeα22dαQx 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 · .
X,X𝒩mσ2:PrX>x=Qxmσ X X m σ 2 X x Q x m σ
(2)
Integrating by parts, Q· Q · is bounded (for x>0 x 0 ) by
12πx1+x2ex22Qx12πxex22 1 2 x 1 x 2 x 2 2 Q x 1 2 x x 2 2
(3)
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 Qx 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 x1 x 1 .
x,x1:122πex22Qx x x 1 1 2 2 x 2 2 Q x
(4)

We will have occasion to evaluate the expected value of eaX+bX2 a X b X 2 where X𝒩mσ2 X m σ 2 and aa, bb are constants. By definition, EeaX+bX2=12πσ2eax+bx2xm22σ2dx 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 (12σ2((12bσ2)x22(m+aσ2)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 (12bσ22σ2xm+aσ212bσ22)+12bσ22σ2m+aσ212bσ22m22σ2 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, EeaX+bX2=e12bσ22σ2m+aσ212bσ22m22σ212πσ2e(12bα22σ2xm+aσ212bσ22)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=xm+aσ212bσ2σ12bσ2 α x m a σ 2 1 2 b σ 2 σ 1 2 b σ 2 which implies that we must require that 12bσ2>0 1 2 b σ 2 0 (or b<12σ2 b 1 2 σ 2 ). We then obtain EeaX+bX2=e12bσ22σ2m+aσ212bσ22m22σ2112bσ212πeα22dα 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

b,b<12σ2:EeaX+bX2=e12bσ22σ2m+aσ212bσ22m22σ212bσ2 b b 1 2 σ 2 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 b σ 2
(5)
Important special cases are
1. a=0 a 0 , X𝒩mσ2 X m σ 2 EebX2=ebm212bσ212bσ2 b X 2 b m 2 1 2 b σ 2 1 2 b σ 2
2. a=0 a 0 , X𝒩0σ2 X 0 σ 2 EebX2=112bσ2 b X 2 1 1 2 b σ 2
3. X𝒩0σ2 X 0 σ 2 EeaX+bX2=ea2σ22×(12bσ2)12bσ2 a X b X 2 a 2 σ 2 2 1 2 b σ 2 1 2 b σ 2

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

pXx=1det(2πK)e(12(xm)TK-1(xm)) p X x 1 2 K 1 2 x m K x m
(6)
If complex-valued, the joint distribution of a circular Gaussian random vector is given by
pXx=1det(πK)e((x m X )T K X -1(x m X )) p X x 1 K x m X K X x m X
(7)
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 =EXXT 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 X𝒩 m X K X X m X K X . After applying a linear transformation to Gaussian random vector, such as Y=AX Y A X , the result is also a Gaussian random vector (a random variable if the matrix is a row vector): Y𝒩A 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 iu=eiuT m X 12uT 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 .

## Footnotes

1. Gaussian random variables are also known as normal random variables.
2. E X 1 X N =( Φ X iu u N ) u 2 u 1 | u =0 X 1 X N u 0 u 1 u 2 u N Φ X u

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