Signal Processing - MATLAB Simulations for Radar Systems Design - page 631

and, thus, the first central moment is zero. The second central moment is called
the variance and is denoted by the symbol

σ 2

,

) 2 σ 2

= ( XX –

(13.67)

Appendix 13B has some common pdf s and their means and variances.

In practice, the random nature of an electrical signal may need to be
described by more than one random variable. In this case, the joint cdf and pdf
functions need to be considered. The joint cdf and pdf for the two random vari-
ables

X and Y are, respectively, defined by

F XY xy ( , ) = Pr X { ≤ x Y ; ≤ y }

(13.68)

2

∂

f XY ( xy , ) = F XY ( xy , )

(13.69)

∂ xy ∂

The marginal cdf s are obtained as follows:

∞
∫

x
∫

F X () = f UV ( uv , ) udv d = F XY ( x ∞ , )

– ∞ – ∞

(13.70)

∞
∫

y
∫

F Y () = f UV ( uv , ) vdd = F XY ∞ y ( , )

– ∞ – ∞

If the two random variables are statistically independent, then the joint cdf s and
pdf s are, respectively, given by

F XY ( xy , ) = F X () F Y ()

(13.71)

f XY ( xy , ) = f X () f Y ()

(13.72)

Let us now consider a case when the two random variables X and Y are

mapped into two new variables U and V through some transformations T 1

and T 2 defined by

U =

V

( XY , )

1

(13.73)

= ( XY , )

2

The joint pdf, , may be computed based on the invariance of proba-
bility under the transformation. One must first compute the matrix of deriva-
tives; then the new joint pdf is computed as

f UV ( uv , )

f UV ( uv , ) = f XY ( xy , ) J (13.74)