The Gaussian Random Variable1.22003/05/142003/08/04 16:19:26.788 GMT-5DonJohnsondhj@rice.eduDonJohnsondhj@rice.eduEileenKrauseerkrause@rice.eduKyleClarksonkclarks@rice.eduElizabethGregorylizzardg@rice.eduKevinDuhkevinduh@rice.eduMariyahPoonawalamariyah@rice.eduJeffreySilvermanjsilv@rice.edu
The random variable X is said to
be a Gaussian random variableGaussian random variables are also known as
normal random variables. if its probability
density function has the form
pXx12σ2xm22σ2
The mean of such a Gaussian random variable is
m and its variance σ2. As a shorthand notation, this information is denoted
by.
xmσ2. The characteristic function
ΦX· of a Gaussian random variable is given by
ΦXumuσ2u22
No closed form expression exists for the probability
distribution function of a Gaussian random variable. For a
zero-mean, unit-variance, Gaussian random variable
01, the probability that it
exceeds the value
x is denoted by Qx.
Xx1PXx12αxα22Qx
The function
Q· is plotted on logarithmic coordinates. Beyond values
of about two, this function decreases quite rapidly. Two
approximations are also shown that correspond to the upper and
lower bounds given by .
A plot of Q· is shown in . When the
Gaussian random variable has non-zero mean and/or non-unit
variance, the probability of it exceeding
x can also be expressed in terms
of Q·.
XXmσ2XxQxmσ
Integrating by parts,
Q· is bounded (for x0) by
12x1x2x22Qx12xx22
As x becomes large, these bounds
approach each other and either can serve as an approximation to
Q·; the upper bound is usually chosen because of its
relative simplicity. The lower bound can be improved; noting
that the term x1x2 decreases for x1 and that Qx increases as x
decreases, the term can be replaced by its value at x1 without affecting the sense of the bound for x1.
xx1122x22Qx
We will have occasion to evaluate the expected value of aXbX2 where
Xmσ2 and
a, b
are constants. By definition,
aXbX212σ2xaxbx2xm22σ2
The argument of the exponential requires manipulation (i.e.,
completing the square) before the integral can be evaluated.
This expression can be written as
12σ212bσ2x22maσ2xm2
Completing the square, this expression can be written
12bσ22σ2xmaσ212bσ2212bσ22σ2maσ212bσ22m22σ2
We are now ready to evaluate the integral. Using this expression,
aXbX212bσ22σ2maσ212bσ22m22σ212σ2x12bα22σ2xmaσ212bσ22
Let
αxmaσ212bσ2σ12bσ2
which implies that we must require that 12bσ20 (or b12σ2). We then obtain
aXbX212bσ22σ2maσ212bσ22m22σ2112bσ212αα22
The integral equals unity, leaving the result
bb12σ2aXbX212bσ22σ2maσ212bσ22m22σ212bσ2
Important special cases are
a0,
Xmσ2bX2bm212bσ212bσ2a0,
X0σ2bX2112bσ2X0σ2aXbX2a2σ2212bσ212bσ2
The real-valued random vector X is said to be a Gaussian
random vector if its joint distribution function has the
form
pXx12K12xmKxm
If complex-valued, the joint distribution of a circular Gaussian
random vector is given by
pXx1KxmXKXxmX
The vector mX denotes the
expected value of the Gaussian random vector and KX its covariance matrix.
mXXKXXXmXmX
As in the univariate case, the Gaussian distribution of a random
vector is denoted by
XmXKX. After applying a linear transformation to Gaussian
random vector, such as YAX, the result is also a Gaussian random vector (a random
variable if the matrix is a row vector):
YAmXAKXA.
The characteristic function of a Gaussian random vector is given
by
ΦXuumX12uKXu
From this formula, the Nth-order moment formula
for jointly distributed Gaussian random variables is easily
derived. X1…XNu0u1u2…uNΦXuX1…XN𝒫N𝒫NX𝒫N(1)X𝒫N(2)…X𝒫N(N-1)X𝒫N(N)N even𝒫N𝒫NX𝒫N(1)X𝒫N(2)X𝒫N(3)…X𝒫N(N-1)X𝒫N(N)N odd
where 𝒫N denotes a permutation of the first
N integers and 𝒫Ni the ith element of the permutation. For example, X1X2X3X4X1X2X3X4X1X3X2X4X1X4X2X3.