Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 258

According to the CLT and using ( 4.121 ) and ( 4.122 ), we have:

2

e ð x m X Þ

1
2 p

2 s X

f X ðxÞ! p ; as N !1: (4.123)

s X

The proof of the CLT can be found, for example, in [LEO94, pp. 287-288].
The CLT can be interpreted as a property of the convolutions ( 4.120 ) involving a
large number of positive functions. This result is independent of the type of the random
variables in the sum, supposing that the participation of each variable in the sum is small
in comparison with the total sum. The value of N in ( 4.119 ) must be a large number in
order to obtain a good approximation to the normal variable. However, if the PDFs are
“reasonably concentrated near mean value,” [PAP65, p. 267], then a close approxima-
tion to ( 4.123 ) is obtained even for moderate values of N . To see more discussion of the
normal approximation for a finite number of N , see [BRE69, pp. 117-119].
The CLT shows why the normal random variables are so important in different
applications. There are many situations in which a random occurrence can be
considered the result of many independent occurrences where the participation of
each occurrence in the total sum is small. For example, the resulting error of a given
measurement can be viewed as the result of many independent random occurrences
where the participation of each of them in the total error is small, and can thus be
considered a normal random variable.
The CLT is especially important in communications, where the noise which is
added in the channel to the signal, is the result of different sources, and can thus be
presented as normal noise.
Similarly, thermal noise is the result of random independent movement of
electrons and can thus be presented as a normal noise.
In some special cases, the CLT can also be applied to dependent random
variables [PEE93, p. 118].
Finally, we must mention that a more adequate name for the CLT would be the
theorem of normal convergence. However, the word “central” is useful to remind us
that the PDF converges to the normal PDF around the center of the mean value and
more errors are expected at the tails of the PDF.

4.6 Jointly Normal Variables

4.6.1 Two Jointly Normal Variables

Consider two normal random variables, X 1 ¼ Nðm 1 ; s 1 Þ and X 2 ¼ Nðm 2 ; s 2 Þ .
The variables X 1 and X 2 are jointly normal if their joint density function is given as:

h i

2

þ ð x 2 m 2 Þ 2
s 2

ðx 1 m 1 Þ

2 r 1 ; 2 ð x 1 m 1 Þð x 2 m 2 Þ
s 1 s 2

1
2 ð 1 r 1 ; 2 Þ

e

1

s 1

f X 1 ;X 2 ðx 1 ; x 2 Þ¼ q

1 r 1 ; 2

;

2 ps 1 s 2

(4.124)