cc [ flags ] -I/usr/local/include file(s) -L/usr/local/lib -lmcw [ ... ] #include <mathcw.h> extern float phif (float x); extern double phi (double x); extern long double phil (long double x); extern __float80 phiw (__float80 x); extern __float128 phiq (__float128 x); extern long_long_double phill (long_long_double x); extern decimal_float phidf (decimal_float x); extern decimal_double phid (decimal_double x); extern decimal_long_double phidl (decimal_long_double x); extern decimal_long_long_double phidll (decimal_long_long_double x);
NB: Functions with prototypes containing underscores in type names may be available only with certain extended compilers.
phi(x) = (1/sqrt(2pi)) integral(t = -infinity:x) exp((-t**2)/2) dt
= 1 - phic(x).
This is the area under the curve of the standard normal distribution
to the
left
of
x,
sometimes called the
lower-tail
area. In mathematical text,
phi(x)
is written with an uppercase Greek phi:
\(*F(x).
If a random variable x has a normal distribution with mean \(*m (mu) and standard deviation \(*s (sigma), then the translated and scaled variable r = (x - \(*m)/\(*s has a standard normal distribution. The probability that x lies in the interval [a, b] is then given by
P(a <= x <= b) = \(*F((b - \(*m)/\(*s) - \(*F((a - \(*m)/\(*s).Similarly, for the standard normal distribution, the probability that r lies in the interval [c, d] is
P(c <= r <= d) = \(*F(d) - \(*F(c).From this, it follows that the probability that a normally-distributed random value x exceeds the mean by at least n standard deviations is therefore given by
P((\(*m + n\(*s) <= x <= \(if) = \(*F(\(if) - \(*F(((\(*m + n\(*s) - \(*m)/\(*s)
= 1 - \(*F(((\(*m + n\(*s) - \(*m)/\(*s)
= \(*F_c(((\(*m + n\(*s) - \(*m)/\(*s)
= \(*F_c(n).
That probability is independent of the mean and standard deviation, and
drops off rapidly, as this table illustrates:
n 0 1 2 3 4 5 6
\(*F_c(n) 0.5 0.159 0.0228 0.00135 3.17e-05 2.87e-07 9.87e-10
These data lead to the handy rules of thumb that only about
one in a thousand
normally-distributed random values lies more than
three
standard deviations from the mean, and fewer than
one in a million
lie more than
five
standard deviations away.
Caution: While the two functions share the simple relation phi(x) + phic(x) = 1, it is only accurate to compute the larger from the smaller.