Random Variable - Random Signals and Processes Primer with MATLAB - page 110

2.8.4 Functions Which Give Moments

It is well known that the Fourier and Laplace transforms are crucial in the description
of properties of deterministic signals and systems. For similar reasons, we can apply
Fourier and Laplace transforms to the PDF, because the PDF is a deterministic
function. Here, we consider the characteristic function and the moment generating
function (MGF). Both functions are very useful in determining the moments of the
random variable and particularly in simplifying the description of the convolution of
probability densities, as we will see later in Chap. 3 .

2.8.4.1 Characteristic Function

The characteristic function of the random variable X denoted as f X ðoÞ is defined as
the expected value of the complex function:

e joX

¼ cos ðoXÞþj sin ðoXÞ;

(2.385)

p .
Using the definition of the expected value of the function of a random variable
( 2.273 ), we have:

where j ¼

1

f X ðoÞ¼Ef e joX e jox f X ðxÞ d x:

g¼ (2.386)

1

Note that the integral in the right side of ( 2.386 ) is the Fourier transform of
density function f X ( x ), (with a reversal in the sign of the exponent). As a conse-
quence, the importance of ( 2.386 ) comes in knowing that the Fourier transform has
many useful properties. Additionally, using the inverse Fourier transformation, we
can find the density function from its characteristic function,

1

1
2 p

f X ðoÞ e jox d o: f X ðxÞ¼ (2.387)

1

From ( 2.385 ), we can easily see that, for o ¼ 0, the characteristic function is
equal to 1.
Similarly,

1 1

f X ðxÞ d x ¼

e jox

j f X ðoÞ j f X ðxÞ d x ¼ 1 : (2.388)

1 1