The integral of a normal distribution

No need to know the antiderivative of a function to compute its integral.

The normal distribution describes random variables that represent the distance from the origin of points randomly chosen around the origin of a plane, such that

1. their coordinates do not depend on the orientation of the reference frame chosen to describe them;
2. their coordinates are independent on each other;
3. and large distances are less likely than small distances.

Its probability density function (pdf) is

Being x a continuous variable, its pdf does not represent the probability of drawing a given x from the sample, which is zero, being infinite the number of possible values of x. Instead, P(x)dx represents the probability that x is between x and x+dx.

The pdf’s expected value is null, and its variance is equal to 1. The shape of the distribution is as follows:

The area under the curve between −1 and +1 is 0.68, that between −2 and +2 is 0.95 and that between −3 and +3 is greater than 0.99. These values can only be obtained by numerically integrating the pdf, that does not have an antiderivative. The values of the cumulative distribution function of the above pdf can be found…