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B–5 Cumulative Distribution Functions and Probability Density Functions 689
CDF is evaluated first by using F(a) = P(x … a) = lim n : q (n x … a > n), where n is the number of
batteries in the whole collection and n xa is the number of batteries in the collection with voltages
less than or equal to a V, where a is a parameter. The CDF that might be obtained is illustrated
in Fig. B–5a. The associated PDF is obtained by taking the derivative of the CDF, as shown in
Fig. B–5b. Note that f(x) exceeds unity for some values of x, but that the area under f(x) is unity
(a PDF property). (You might check to see that the other CDF and PDF properties are also satisfied.)
THEOREM.
F(b) - F(a) = P(x … b) - P(x … a) = P(a 6 x … b)
= lim
e:0
e 7 0
b+e
c f(x)dxd
L b+e
(B–18)
1.0
0.8
0.6
F (a)
0.4
0.2
0.5 1.0
1.5 2.0
a (volts)
(a) Cumulative Distribution Function
2.0
1.5
1.0
0.5
f (x)
Area = 0.19
Total area = 1.0
0.5 1.0 1.5 2.0
x (volts)
(b) Probability Density Function
Figure B–5
CDF and PDF for a continuous distribution (Example B–3).