The cumulative distribution function (CDF) of the standard normal distribution: error function, or erf(x)

Cumulative distribution function[edit]

The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter  (phi), is the integral

In statistics one often uses the related error function, or erf(x), defined as the probability of a random variable with normal distribution of mean 0 and variance 1/2 falling in the range ; that is

These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions. However, many numerical approximations are known; see below.

The two functions are closely related, namely

For a generic normal distribution f with mean μ and deviation σ, the cumulative distribution function is

The complement of the standard normal CDF, , is often called the Q-function, especially in engineering texts.^[16][17] It gives the probability that the value of a standard normal random variable X will exceed x. Other definitions of the Q-function, all of which are simple transformations of , are also used occasionally.^[18]

The graph of the standard normal CDF  has 2-fold rotational symmetry around the point (0,1/2); that is, . Its antiderivative (indefinite integral)  is .

• The cumulative distribution function (CDF) of the standard normal distribution can be expanded by Integration by parts into a series:

where  denotes the double factorial. As an example, the following Pascal function approximates the CDF:

function CDF(x:extended):extended;
var value,sum:extended;
    i:integer;
begin
  sum:=x;
  value:=x;
  for i:=1 to 100 do
    begin
      value:=(value*x*x/(2*i+1));
      sum:=sum+value;
    end;
  result:=0.5+(sum/sqrt(2*pi))*exp(-(x*x)/2);
end;