Normal distribution
- Don't forget about Standard normal distribution! Alec (talk) 01:30, 14 December 2017 (UTC)
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
Notes:
The MDM of [ilmath]X\sim\text{Nor}(0,\sigma^2)[/ilmath] is [math]\sqrt{\frac{2\sigma^2}{\pi} } [/math][1] , so is related the standard deviation linearly. It's also unaffected by the mean of the distribution - this hasn't been proved but is "obvious" and also verified experimentally.
References
- ↑ From a friend's memory. It has been experimentally confirmed though and is at the very worst an extremely close approximation (on the order of [ilmath]10^{-10} [/ilmath])
