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694
Probability and Random Variables
Appendix B
However, you might ask, “What is the significance of the mean, variance, and other moments
in electrical engineering problems?” In Chapter 6, it is shown that if x represents a voltage or
current waveform, the mean gives the DC value of the waveform. The second moment (r = 2)
taken about the origin (x 0 = 0), which is x 2 , gives the normalized power. s 2 gives the normalized
power in the corresponding AC coupled waveform. Consequently, 3x 2 is the RMS value of
the waveform, and s is the RMS value of the corresponding ac coupled waveform.
In statistical terms, m gives the center of gravity of the PDF, and s gives us the spread
of the PDF about this center of gravity. For example, in Fig. B–5 the voltage distribution for a
collection of flashlight batteries is given. The mean is xq = 1.25 V, and the standard deviation
is s = 0.25 V. This figure illustrates a Gaussian distribution that will be studied in detail in
Sec. B–7. For this Gaussian distribution, the area under f(x) from x = 1.0 to 1.5 V, which
corresponds to the interval xq ; s, is 0.68. Thus, we conclude that 68% of the batteries have
voltages within one standard deviation of the mean value (Gaussian distribution).
There are several ways to specify a number that is used to describe the typical, or
most common, value of x. The mean m is one such measure that gives the center of gravity.
Another measure is the median, which corresponds to the value x = a, where
A third measure is called the mode, which corresponds to the value of x where f(x) is a maximum,
assuming that the PDF has only one maximum. For the Gaussian distribution, all
these measures give the same number, namely, x = m. For other types of distributions, the
values obtained for the mean, median, and mode will usually be nearly the same number.
The variance is also related to the second moment about the origin and the mean, as
described by the following theorem:
THEOREM.
s 2 = x 2 - (xq) 2 F(a) = 1 2 .
A proof of this theorem illustrates how the ensemble average operator notation is used.
Proof.
s 2 = (x - xq) 2
(B–29)
= [x 2 ] - [2xxq] + [(xq)] 2
(B–30)
Because [ # ] is a linear operator, [2xxq] = 2xqxq = 2(xq) 2 . Moreover, (xq) 2 is a constant, and the
average value of a constant is the constant itself. That is, for the constant c,
q
q
cq = cf(x) dx = c f(x) dx = c
L L
-q
-q
(B–31)
So, using Eq. (B–31) in Eq. (B–30), we obtain
which is equivalent to Eq. (B–29).
s 2 = x 2 - 2(xq) 2 + (xq) 2