Normal distribution

Definition

The normal distribution has a Probability density function or PDF, [ilmath]f:\mathbb{R}\rightarrow\mathbb{R} [/ilmath] given by: [math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]

• [math]f(x):=\frac{1}{\sigma\sqrt{2\pi} } e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma} \right)^2}[/math]

The Cumulative density function or CDF is naturally given by:

• [math]F(x):=P(-\infty < X < t)=\frac{1}{\sigma\sqrt{2\pi} }\int^t_\infty e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma} \right)^2}\d x[/math]

In this definition:

• [ilmath]\sigma[/ilmath] is the standard deviation of the distribution (so [ilmath]\sigma^2[/ilmath] is the variance) and
• [ilmath]\mu[/ilmath] is the mean