Random Variable - Random Signals and Processes Primer with MATLAB - page 45

This density is shown in Fig. 2.14b .
The probability that the variable is less than 4 corresponds to the following
integral of the PDF

ð

4

ð

4

1
4 d x ¼ 1

PfX 4 g¼ f X ðxÞ d x ¼ = 2 : (2.66)

2 2

Note that this value corresponds to the area under the density, as shown in
Fig. 2.14b , and alternately to the only one point F X (4) ¼ 1/2 at the distribution
function in Fig. 2.14a .

2.3.2 Delta Function

As mentioned in Sect. 2.1 , the distribution of a discrete random variable is in a
stair-step form. In order to circumvent the problem in defining the derivative,
we use the unit-impulse function d ( x ), also called the delta function ,

d u ð x Þ
d x :

dðxÞ¼ (2.67)

From here

ð

x

uðxÞ¼ dðvÞ d v: (2.68)

1

A delta function d ( x ) is a continuous function of x which is defined in the range
½1;1 . It can be interpreted as a function with infinite amplitude at x ¼ 0 and
zero duration with unitary area,

8
<

1

1 for x ¼ 0

dðxÞ¼ dðxÞ d x ¼ 1 (2.69)

otherwise ;

:

0

1

We usually present the delta function as an arrow occurring at x ¼ 0 and with
unity magnitude as shown in Fig. 2.15a . The interpretation of the delta function is
shown in Fig. 2.15b , demonstrating that it has zero duration and infinite amplitude;
but an area equal to 1, assuming

Δ x ! 0.
A general expression is obtained if we shift the delta function d ( x )by x 0 ,

8
<

1

1 for x ¼ x 0 ;

dðx x 0 Þ¼ dðx x 0 Þ d x ¼ 1

otherwise ;

(2.70)

:

0

1