m
x
Ex
I
ÀI
Xf XdX
1
X
2
À X
1
X
2
X
1
XdX
X
2
X
1
2
:
3:3:13
The variance of the function is
s
2
x
Ex
2
À m
2
x
I
ÀI
X
2
f XdX À m
2
x
1
X
2
À X
1
X
2
X
1
X
2
dX À m
2
x
1
X
2
À X
1
:
X
3
2
À X
3
1
3
À m
2
x
X
2
2
X
1
X
2
X
2
1
3
À
X
2
X
1
2
4
X
2
À X
1
2
12
:
3:3:14
In general, if x is a uniformly distributed random variable in the interval (ÀD, D), the
mean is 0 (m
x
0) and the variance is s
2
x
D
2
=3.
Example 3.3: The MATLAB function rand generates pseudo-random numbers
uniformly distributed in the interval (0, 1). From Equation (3.3.13), the mean of
the generated pseudo-random numbers is 0.5. From (3.3.14), the variance is
computed as 1/12. To generate zero-mean random numbers, we subtract 0.5
from every generated number. The numbers are now distributed in the interval
(À0.5, 0.5). To make these pseudo-random numbers with unit-variance, i.e.,
s
2
x
D
2
=3 1, the generated numbers must be equally distributed in the interval
(À
3
p
,
3
p
). Therefore we have to multiply 2
3
p
to every generated number that
was subtracted by 0.5. The following MATLAB statement can be used to generate
the uniformly distributed random numbers with mean 0 and variance 1:
xn 2*sqrt(3)*(rand À 0:5);
For two random variables x and y, we have
Ex y Ex Ey,
3:3:15
i.e., the mean value of the sum of random variables equals the sum of mean values. The
correlation of x and y is denoted as E[xy]. In general, Exy T Ex Á Ey. However, if x
and y are uncorrelated, then the correlation can be written in the form
Exy ExÁEy:
3:3:16
Statistical independence of x and y is sufficient to guarantee that they are uncorrelated.
If the random variables x
i
are independent with the mean m
i
and variance s
2
i
, the
random variable y is defined as
y x
1
x
2
Á Á Á x
N
N
i1
x
i
:
3:3:17
94
DSP FUNDAMENTALS AND IMPLEMENTATION CONSIDERATIONS