Difference between revisions of "Normal distribution"
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| + | {{Stub page|grade=A*|msg=In development! [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 01:30, 14 December 2017 (UTC) | ||
| + | * Don't forget about [[Standard normal distribution]]! [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 01:30, 14 December 2017 (UTC) }} | ||
==Definition== | ==Definition== | ||
The normal distribution has a [[Probability density function]] or [[PDF]], {{M|f:\mathbb{R}\rightarrow\mathbb{R} }} given by: {{Extra Maths}} | The normal distribution has a [[Probability density function]] or [[PDF]], {{M|f:\mathbb{R}\rightarrow\mathbb{R} }} given by: {{Extra Maths}} | ||
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* {{M|\sigma}} is the [[standard deviation]] of the distribution (so {{M|\sigma^2}} is the [[variance]]) and | * {{M|\sigma}} is the [[standard deviation]] of the distribution (so {{M|\sigma^2}} is the [[variance]]) and | ||
* {{M|\mu}} is the [[mean]] | * {{M|\mu}} is the [[mean]] | ||
| + | ==Notes:== | ||
| + | The [[MDM]] of {{M|X\sim\text{Nor}(0,\sigma^2)}} is {{MM|\sqrt{\frac{2\sigma^2}{\pi} } }}<ref>From a friend's memory. It has been [[experimental confirmation|experimentally confirmed]] though and is at the very worst an extremely close approximation (on the order of {{M|10^{-10} }})</ref> , so is related the {{link|standard deviation|distribution}} linearly. It's also unaffected by the mean of the distribution - this hasn't been proved but is "obvious" and also verified experimentally. | ||
| + | |||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Statistics}} | {{Definition|Statistics}} | ||
Revision as of 01:30, 14 December 2017
- 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])
