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688
Probability and Random Variables
Appendix B
Properties of CDFs and PDFs
Some properties of the CDF are as follows:
1. F(a) is a nondecreasing function.
2. F (a) is right-hand continuous. That is,
3.
4. 0 … F(a) … 1.
5. F(-q) = 0.
6. F(+q) = 1.
lim F(a + e) = F(a)
e:0
e 7 0
a+e
F(a) = lim
e:0
e 7 0 L f(x)dx
- q
(B–16)
Note that the e is needed to account for a discrete point that might occur at x = a. If there is no
discrete point at x = a, the limit is not necessary.
Some properties of the PDF are as follows:
1. f(x) 0. That is, f(x) is a nonnegative function.
q
2. f(x) dx = F(+q) = 1.
(B–17)
L-q
As we will see later, f(x) may have values larger than unity; however, the area under f(x) is
equal to unity. These properties of the CDF and PDF are very useful in checking results of
problems. That is, if a CDF or a PDF violates any of these properties, you know that an error
has been made in the calculations.
Discrete and Continuous Distributions
Example B–2 is an example of a discrete, or point, distribution. That is, the random variable
has M discrete values x 1 , x 2 , x 3 ,..., x M . (M = 7 in this example.) Consequently, the CDF
increased only in jumps [i.e., F(a) was discontinuous] as a increased, and the PDF consisted
of delta functions located at the discrete values of the random variable. In contrast to this
example of a discrete distribution, there are continuous distributions, one of which is illustrated
in the next example. If a random variable is allowed to take on any value in some interval,
it is a continuously distributed random variable in that interval.
Example B–3 ACONTINUOUS DISTRIBUTION
Let the random variable denote the voltages that are associated with a collection of a large number
of flashlight batteries (1.5-V cells). If the number of batteries in the collection were infinite, the
number of different voltage values (events) that we could obtain would be infinite, so that the
distributions (PDF and CDF) would be continuous functions. Suppose that, by measurement, the