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# Fundamental Probability Density Functions and Properties

Module by: Clayton Scott, Robert Nowak. E-mail the authors

## Gaussian (Normal)

The Gaussian probability density function (PDF) (also referred to as the normal PDF) for a scalar random variable xx is defined as

x ,x :px=12πσ2e(12σ2xμ2) x x p x 1 2 σ 2 1 2 σ 2 x μ 2
(1)
where μμ is the mean and σ2 σ 2 is the variance of xx. It is denoted by 𝒩μσ2 μ σ 2 and we say that x𝒩μσ2 x μ σ 2 , where "∼" means "is distributed according to."

The cumulative distribution function (CDF) for μ=0 μ 0 and σ2=1 σ 2 1 , for which the PDF is termed a standard normal PDF, is defined as

Φx=x12πe(12t2)d t Φ x t x 1 2 1 2 t 2
(2)
A more convenient description, which is termed the right-tail probability and is the probability of exceeding a given value, is defined as Qx=1Φx Q x 1 Φ x , where
Qx=x12πe(12t2)d t Q x t x 1 2 1 2 t 2
(3)
An approximation that is sometimes useful is
Qx12πxe(12x2) Q x 1 2 x 1 2 x 2
(4)

It is shown in Figure 1 along with the exact value of Qx Q x . The approximation is quite accurate for x>4 x 4 .

The multivariate Gaussian PDF of an n × 1 n×1 random vector x x is defined as

px=12πn2detCe(12(xμ)TC-1(xμ)) p x 1 2 n 2 C 1 2 x μ C x μ
(5)
where μμ is the mean vector and CC is the covariance matrix and is denoted by 𝒩μC μ C . It is assumed that CC is positive definite and hence C-1 C exists. The mean vector is defined as
i :μi=E x i i i 1 2 n μ i x i
(6)
and the covariance matrix as
i j ,i12nj12n:Ci,j=E( x i E x i )( x j E x j ) i j i 1 2 n j 1 2 n C i j x i x i x j x j
(7)
or in more compact form
C=E(xEx)(xEx)T C x x x x
(8)

## Chi-Squared (Central)

A chi-squared PDF with νν degrees of freedom is defined as

px={12ν2Γν2xν21e(12x)  if  x>00  if  x<0 p x 1 2 ν 2 Γ ν 2 x ν 2 1 1 2 x x 0 0 x 0
(9)
and is denoted by χ ν 2 χ ν 2 . The degrees of freedom νν is assumed to be an integer with ν1 ν 1 . The function Γu Γ u is the Gamma function which is defined as
Γu=0tu1etd t Γ u t 0 t u 1 t
(10)
The relations Γu=(u1)Γu1 Γ u u 1 Γ u 1 for any uu, Γ12=π Γ 1 2 , and Γn=(n1)! Γ n n 1 for nn an integer.

The chi-squared PDF arises as the PDF of xx where x= i =1ν x i 2 x i 1 ν x i 2 if x i 𝒩01 x i 0 1 and the x i x i 's are independent and identically distributed (IID). By the latter we mean that each x i x i is independent of the others and each x i x i has the same PDF (identically distributed). The mean and variance are

Ex=ν x ν
(11)
σ(x)2=2ν x 2 ν
(12)
A specific case of interest occurs when ν=2 ν 2 so that
px={12e(12x)  if  x>00  if  x<0 p x 1 2 1 2 x x 0 0 x 0
(13)
and is referred to as an exponential PDF (Figure 2).

## F (Central)

The F PDF arises as the ratio of two independent χ2 χ 2 random variables. Specifically, if

x= x 1 ν 1 x 2 ν 2 x x 1 ν 1 x 2 ν 2
(14)
where x 1 χ ν 1 2 x 1 χ ν 1 2 , x 2 χ ν 2 2 x 2 χ ν 2 2 , and x 1 x 1 and x 2 x 2 are independent, then xx has the F PDF. It is given by
px={ ν 1 ν 2 ν 1 2B ν 1 2 ν 2 2x ν 1 211+ ν 1 ν 2 x ν 1 + ν 2 2  if  x>00  if  x<0 p x ν 1 ν 2 ν 1 2 B ν 1 2 ν 2 2 x ν 1 2 1 1 ν 1 ν 2 x ν 1 ν 2 2 x 0 0 x 0
(15)
where Buv B u v is the Beta function, which can be related to the Gamma functions as
Buv=ΓuΓvΓu+v B u v Γ u Γ v Γ u v
(16)
The PDF is denoted by F ν 1 , ν 2 F ν 1 , ν 2 as an F PDF with ν 1 ν 1 numerator degrees of freedom and ν 2 ν 2 denominator degrees of freedom.

The mean and variance are

ν 2 , ν 2 >2:Ex= ν 2 ν 2 2 ν 2 ν 2 2 x ν 2 ν 2 2
(17)
ν 2 , ν 2 >4:σ(x)2=2 ν 2 2( ν 1 + ν 2 2) ν 1 ν 2 22( ν 2 4) ν 2 ν 2 4 x 2 ν 2 2 ν 1 ν 2 2 ν 1 ν 2 2 2 ν 2 4
(18)

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