Definition of the normal distribution

Many measured quantities in the natural sciences follow a normal distribution, for example heights, masses, ages, random errors, I. Q. scores, examination results and the normal distribution is one of the most important continuous distributions.

The curve that represents the normal distribution is the famous bell-shaped curve that is extensively used like in the representation of I.Q. scores. The average I.Q. score is \(100\).

In general, the normal curve of a normal random variable \(X\) is characterised by the two parameters of \(X\), namely the mean \(\mu\) and variance \(\sigma^2\). In symbol : \(X \sim N(\mu, \sigma^2)\). Note that \(X\) is a continuous random variable.

The probability density function (pdf) for the normal distribution is given by

\[\mathrm{f}(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]

The function does not have an indefinite integral or antiderivative that can expressed as elementary functions, and therefore, in most cases, the use of the GDC is required in order to calculate the required probabilities.

Despite this, it is possible to show using bivariate calculus, namely Fubini's Theorem, that the area under the normal distribution curve over the real line is indeed equal to unity (i.e., \(1\)).