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B–9 Multivariate Statistics 715
Central Limit Theorem
If we have the sum of a number of independent random variables with arbitrary one-dimensional
PDFs, the central limit theorem states that the PDF for the sum of these independent random
variables approaches a Gaussian (normal) distribution under very general conditions. Strictly
speaking, the central limit theorem does not hold for the PDF if the independent random variables
are discretely distributed. In this case, the PDF for the sum will consist of delta functions (not
Gaussian, which is continuous); however, if the delta functions are “smeared out” (e.g., if the
delta functions are replaced with rectangles that have corresponding areas), the resulting PDF
will be approximately Gaussian. Regardless, the cumulative distribution function (CDF) for the
sum will approach that of a Gaussian CDF.
The central limit theorem is illustrated in the following example.
Example B–12 PDF FOR THE SUM OF THREE INDEPENDENT,
UNIFORMLY DISTRIBUTED RANDOM VARIABLES
The central limit theorem will be illustrated by evaluating the exact PDF for the sum of three
independent uniformly distributed random variables. This exact result will be compared with the
Gaussian PDF, as predicted by the central limit theorem.
Let each of the independent random variables x i have a uniform distribution, as shown in
Fig. B–14a. The PDF for y 1 = x 1 + x 2 , denoted by f(y 1 ), is obtained by the convolution operation
described by Eq. (B–108) and Fig. 2–7. This result is shown in Fig. B–14b. It is seen that after
only one convolution operation the PDF for the sum (which is a triangle) is going toward the
Gaussian shape. The PDF for y 2 = x 1 + x 2 + x 3 , f(y 2 ), is obtained by convolving the triangular PDF
with another uniform PDF. The result is
0, y
3
2 … -
2 A
1
2A 3 a 3 2
2 A + y 3
2b , -
2 A … y 2 …- 1 2 A
1
f(y 2 ) = i
(B–109)
2A 3 a 3 2 A2 - 2y 2 2 b, ƒ y 2 ƒ … 1 2 A
1
2A 3 a 3 2 A - y 1
2b,
2 A … y 2 … 3 2 A
0, y 2 Ú 3 2 A
This curve is plotted by the solid line in Fig. B–14c. For comparison purposes, a matching
Gaussian curve, with 1>(12p s) = 3>(4A), is also shown by the dashed line. It is seen that f(y 2 )
is very close to the Gaussian curve for ƒ y 2 ƒ 6 3 2 A, as predicted by the central limit
theorem. Of course, f(y 2) is certainly not Gaussian for ƒ y 2 ƒ 7 3 because K
2 A, f(y 2 ) 0 in this
region, whereas the Gaussian curve is not zero except at y = ; q. Thus, we observe that the
Gaussian approximation (as predicted by the central limit theorem) is not very good on the tails
of the distribution. In Chapter 7, it is shown that the probability of bit error for digital systems is
obtained by evaluating the area under the tails of a distribution. If the distribution is not known to
be Gaussian and the central limit theorem is used to approximate the distribution by a Gaussian
PDF, the results are often not very accurate, as shown by this example. However, if the area under
the distribution is needed near the mean value, the Gaussian approximation may be very good.
See ExampleB_12.m for plots of Fig. B–14c for a selected value of A.