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.
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).phi(x) = (1/sqrt(2pi)) integral(t = -infinity:x) exp((-t**2)/2) dt = 1 - phic(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
That probability is independent of the mean and standard deviation, and drops off rapidly, as this table illustrates: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).
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.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
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.