This module introduces approximation formulae for the Gaussian error Integral
A Gaussian pdf with unit variance is given by:
The probability that a signal with a pdf given by
lies above a given threshold
is given by the Gaussian Error
Integral or
function:
There is no analytical solution to this integral, but it has asimple relationship to the error function,
, or its complement,
, which are tabulated in many books of mathematical
tables.
and
Therefore,
Note that
and
, and therefore
and
very rapidly as
becomes large.
It is useful to derive simple approximations to
which can be used on a calculator and avoid the need
for tables.
Let
, then:
Now if
, we may obtain an approximate solution by replacing
the
term in the integral by unity, since it will initially
decay much slower than the
term. Therefore
This approximation is an upper bound, and its ratio to the true
value of
becomes less than
only when
, as shown in
. We
may obtain a much better approximation to
by altering the denominator above from (
) to (
) to give:
This improved approximation gives a curve indistinguishable from
in
and its ratio
to the true
is now within
of unity for all
as shown in
. This
accuracy is sufficient for nearly all practical problems.
and the simple approximation of
.The ration of the improved approximation of
in
to the true
value, obtained by numerical integration.
