Random Variable - Random Signals and Processes Primer with MATLAB - page 46

This function is presented in Fig. 2.15c . Taking the delayed delta function we
can write:

d u ð x x 0 Þ
d x

dðx x 0 Þ¼ : (2.71)

For any function g ( x ), continuous at the point x 0 , we have:

1

gðxÞdðx x 0 Þ d x ¼ gðx 0 Þ: (2.72)

1

The interpretation of ( 2.72 ) is given in Fig. 2.15d .

2.3.3 Densities of Discrete and Mixed Random Variables

From ( 2.60 ) and the distribution of the discrete random variable ( 2.24 ), we arrive at
the density of a discrete random variable in the form:

!

1

¼ 1
i¼1

d
d x

d
d x

f X ðxÞ¼ Pfx i guðx x i Þ Pfx i g ð uðx x i Þ Þ: (2.73)

i¼1

Fig. 2.15 Delta function: definition and characteristics