563489578934

700
Probability and Random Variables
Appendix B
This function is still properly normalized, as demonstrated by Eq. (B–49). First, we need to
show that the parameter a is the mean:
q
q
1
m = x f(x) dx =
xe -(x-a)2 >(2b 2 )
dx (B–51)
L -q
12p b L-q
Making a change in variable, we let y = (x - a)>b; then,
q
1
m =
(by + a) e -y2 >2 dy
12p L-q
or
q
q
b
m =
(ye -y2 >2 1
) dy + aa >2 (B–52)
12p L-q
L -q 12p e-y2 dyb
The first integral on the right of Eq. (B–52) is zero, because the integrand is an odd function
and the integral is evaluated over symmetrical limits. The second integral on the right is
the integral of a properly normalized Gaussian PDF, so the integral has a value of unity. Thus,
Eq. (B–52) becomes
m = a
(B–53)
and we have shown that the parameter a is the mean value.
The variance is
q
s 2 = (x - m) 2 f(x) dx
L
-q
1
=
(x - m) 2 e -(x-m)2 >(2b 2 )
dx
12p b L-q
Similarly, we need to show that s 2 = b 2 . This will be left as a homework exercise.
q
Example B–6 PLOTTING THE PDF FOR A GAUSSIAN RANDOM VARIABLE
Write a MATLAB program that will ask for values for m and s and then plot the corresponding
Gaussian PDF. See ExampleB_06.m for the solution.
(B–54)
The next question to answer is, “What is the CDF for the Gaussian distribution?”
THEOREM. The cumulative distribution function (CDF) for the Gaussian distribution is
F(a) = Qa m - a b = 1 2
(B–55)
s
erfca m - a
12 s b
where the Q function is defined by
Q(z) ! 1
q
e -l2 >2 dl
(B–56)
12p Lz
and the complementary error function (erfc) is defined as
erfc(z) ! 2
1p Lz
q
e -l2 dl
(B–57)