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B–5 Cumulative Distribution Functions and Probability Density Functions 691
L is the largest integer such that x L … a, L … M, and M is the number of points in the
discrete distribution. Here it is assumed that the discrete points x i are indexed so that
they occur in ascending order of the index. That is, x 1 6 x 2 6 x 3 6 ... 6 x M .
This theorem was illustrated in Fig. B–3, which is a plot of Eq. (B–20) for Example B–2.
In electrical engineering problems, the CDFs or PDFs of waveforms are relatively
easy to obtain by using the relative frequency approach, as described by Eqs. (B–14) and
(B–15). For example, the PDFs for triangular and square waves are given in Fig. B–6.
These are obtained by sweeping a narrow horizontal window, ¢ x volts wide, vertically
across the waveforms and measuring the relative frequency of occurrence of voltages in the
¢ x window. The time axis is divided into n intervals, and the waveform appears n ∆x times
in these intervals in the ¢ x window. A rough idea of the PDF of a waveform can also be
obtained by looking at the waveform on an analog oscilloscope. Using no horizontal sweep,
the intensity of the presentation as a function of the voltage (y-axis) gives the PDF. (This
assumes that the intensity of the image is proportional to the time that the waveform dwells
in the window ¢ y units wide.) The PDF is proportional to the intensity as a function of y
(the random variable).
x (t)
f (x)
A
f (x)
x
A
1
2A
t
A
A
x
A
x
A
(a) Triangular Waveform and Its Associated PDF
x (t)
f (x)
A
x
A
f (x)
A
t
A
Weights of
d functions
are 0.5
x
(b) Square Waveform and Its Associated PDF
Figure B–6
PDFs for triangular waves and square waves.