Fundamental Generation Systems

Appendix 5Normal Distribution

A real random variable X follows a normal or Gaussian distribution of mean u and standard deviation σ if its probability density f is given by:

This function is determined by knowing the parameters μ and σ. We note that X ~ N(μ,σ): X follows the law N(μ,σ).

The graphical representation of f(x) leads to the famous "bell curve".

Regardless of the values of the parameters, the Gaussian distribution has a maximum amplitude known as the statistical mode. The mean u corresponds to the abscissa of this maximum amplitude. The intervals that are typically considered are [μ-σ,μ+σ], [μ-2σ,μ+2σ], [μ-3σ,μ+3σ] and [μ-4σ,μ+4σ], which respectively contain 68.3, 95.5, 99.7 and 99.99% of the considered statistical population.

The distribution function F is defined by:

Figure A5.1. Influence of the standard deviation at a constant mean. Probability density of the Gaussian distribution – mean of 5

Figure A5.2. Influence of the standard deviation at a constant mean. Distribution function of the Gaussian distribution – mean of 5

Standard normal distribution: a Gaussian ...