563489578934

690
Probability and Random Variables
Appendix B
Proof.
F(b) - F(a) = lim
e:0
e 7 0
a+e
b+e
= lim c f(x) dx + f(x) dx
L L a+e L
e:0
e 7 0
b+e
= lim c f(x) dxd
L
e:0
e 7 0
Example B–3 (Continued)
c
L
b+e
- q
-q
a+e
a+e
f(x) dx - f(x) dxd
L
-q
a+e
- q
f(x) dxd
Suppose that we wanted to calculate the probability of obtaining a battery having a voltage
between 1.4 and 1.6 V. Using this theorem and Fig. B–5, we compute
1.6
P(1.4 6 x … 1.6) = f(x) dx = F(1.6) - F(1.4) = 0.19
L
1.4
We also realize that the probability of obtaining a 1.5-V battery is zero. Why? However, the probability
of obtaining a 1.5 V ; 0.1 V battery is 0.19.
Example B–4 PDF AND CDF FOR A GAUSSIAN RANDOM VARIABLE
Let x be a Gaussian RV with mean m and variance s. Using MATLAB, find the values for m and
s in order to give the results shown in Fig. B–5. Plot your results and compare with
Fig. B–5. Using MATLAB, evaluate P(1.4 6 x … 1.6). See ExampleB_04.m for the solution.
In communication systems, there are digital signals that may have discrete distributions
(one discrete value for each permitted level in a multilevel signal), and there are analog
signals and noise that have continuous distributions. There may also be mixed distributions,
which contain discrete, as well as continuous, values. For example, these occur when a continuous
signal and noise are clipped by an amplifier that is driven into saturation.
THEOREM.
If x is discretely distributed, then
M
f(x) = a P(x i ) d (x - x i )
i=1
(B–19)
where M is the number of discrete events and P(x i ) is the probability of obtaining the
discrete event x i .
This theorem was illustrated by Example B–2, where the PDF of this discrete distribution
is plotted in Fig. B–4.
THEOREM. If x is discretely distributed, then †
L
F(a) = a P(x i )
(B–20)
† M
Equation (B–20) is identical to F(a) = a i = 1 P(x 1)u(a - x i ), where u(y) is a unit step function
[defined in Eq. (2–49)].
i=1