Difference between revisions of "Normal distribution"
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| − | The '''normal distribution''' is | + | {|align="right" border="1" |
| − | The normal probability density function (PDF) is | + | |- |
| + | |<center>probability density function</center> | ||
| + | |- | ||
| + | |[[Image:Norm.png|px=200]] | ||
| + | |- | ||
| + | |<center>cumulative probability function</center> | ||
| + | |- | ||
| + | |[[Image:Law-norm.png|px=200]] | ||
| + | |- | ||
| + | |} | ||
| + | The '''normal distribution''' is a key distribution in the field of [[probability]]. It is also known as the Gaussian distribution, after [[mathematician]] [[Carl Friedrich Gauss|Carl Gauss]], and the [[bell curve]]. | ||
| + | The normal [[probability density function]] (PDF) is | ||
:<math> | :<math> | ||
| − | + | f(x)=\frac{1}{\sqrt{2\pi\sigma^2}} | |
\exp\left[-\frac{1}{2\sigma^2}\left(x-\mu\right)^2\right] | \exp\left[-\frac{1}{2\sigma^2}\left(x-\mu\right)^2\right] | ||
</math> | </math> | ||
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<math>\mu</math> is the [[mean]] and <math>\sigma^2</math> is the [[variance]]. | <math>\mu</math> is the [[mean]] and <math>\sigma^2</math> is the [[variance]]. | ||
| − | [[ | + | If <math>\mu=0</math> and <math>\sigma=1</math>, the distribution is called the ''standard normal distribution'', often denoted by <math>\phi</math>: |
| + | :<math> | ||
| + | \phi(x) = \frac{1}{\sqrt{2\pi}} | ||
| + | \exp(-\frac{1}{2} x^2). | ||
| + | </math> | ||
| + | |||
| + | |||
| + | == See also == | ||
| + | *[[Central limit theorem]] | ||
| + | [[Category:Probability and Statistics]] | ||
Latest revision as of 16:30, July 29, 2016
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The normal distribution is a key distribution in the field of probability. It is also known as the Gaussian distribution, after mathematician Carl Gauss, and the bell curve. The normal probability density function (PDF) is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\frac{1}{\sqrt{2\pi\sigma^2}} \exp\left[-\frac{1}{2\sigma^2}\left(x-\mu\right)^2\right] }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is the mean and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2} is the variance.
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu=0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma=1} , the distribution is called the standard normal distribution, often denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} :
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = \frac{1}{\sqrt{2\pi}} \exp(-\frac{1}{2} x^2). }