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B–7 Examples of Important Distributions 701
It can also be shown that
erfc(z) = 1 - erf(z)
where the error function is defined as
(B–58)
erf(z) ! 2 e -l2 dl
(B–59)
1p L0
The Q function and the complementary error function, as used in Eq. (B–55), give the
same curve for F(a). Because neither of the corresponding integrals, given by Eqs. (B–56) and
(B–57), can be evaluated in closed form, math tables (see Sec. A–10, Appendix A), numerical
integration techniques, or closed-form approximations must be used to evaluate them. The
equivalence between the two functions is
z
Q(z) = 1 2 erfca z 12 b
(B–60)
Communication engineers often prefer to use the Q function instead of the erfc
function, since solutions to problems written in terms of the Q function do not require the
1
writing of the
2
and 1> 12 factors. Conversely, the advantage of using the erf(z) or erfc(z) functions
is that they are one of the standard functions in MATLAB and these functions are also
available on some hand calculators. However, textbooks in probability and statistics usually
give a tabulation of the normalized CDF. This is a tabulation of F(a) for the case of m = 0 and
s = 1, and it is equivalent to Q(-a) 1
1
and
2 erfc(-a> 12). Since Q(z) and
2 erfc(z> 12)
are equivalent, which one is used is a matter of personal preference. We use the Q-function
notation in this book.
Proof. Proof of a theorem for the Gaussian CDF
a
a
1
F(a) = f(x) dx =
e -(x-m)2 >(2s 2) dx (B–61)
L -q 12p s L-q Making a change in variable, let y = (m - x)>s:
1 (m-a)>s
F(a) =
e -y2 >2
(-sdy)
12p s Lq
or
q
1
F(a) =
e -y2 >2 dy = Qa m - a
12p L(m-a)>s
s
b
Similarly, F(a) may be expressed in terms of the complementary error function.
(B–62)
(B–63)
Example B–7 PLOTTING THE CDF FOR A GAUSSIAN RANDOM VARIABLE
Write a MATLAB program that will ask for values for m and s and then plot the corresponding
Gaussian CDF. See ExampleB_07.m for the solution.