Difference between revisions of "Normal distribution"

Revision as of 12:22, 24 November 2015 (view source) Alec (Talk | contribs) (Created page with "==Definition== The normal distribution has a Probability density function or PDF, {{M|f:\mathbb{R}\rightarrow\mathbb{R} }} given by: {{Extra Maths}} * {{MM|1=f(x):=\fr...")		Revision as of 01:30, 14 December 2017 (view source) Alec (Talk | contribs) m (Adding some stuff) Newer edit →
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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}}

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Revision as of 01:30, 14 December 2017

Definition

The normal distribution has a Probability density function or PDF, $f:\mathbb{R}\rightarrow\mathbb{R}$ given by:

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$

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

The Cumulative density function or CDF is naturally given by:

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

In this definition:

• $\sigma$ is the standard deviation of the distribution (so $\sigma^2$ is the variance) and
• $\mu$ is the mean

Notes:

The MDM of $X\sim\text{Nor}(0,\sigma^2)$ is

$$\sqrt{\frac{2\sigma^2}{\pi} }$$ ^[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

1. Jump up ↑ 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 $10^{-10}$)