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Normal distribution

In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f ( x ) = 1 2 π σ 2 exp ⁡ ( − ( x − μ ) 2 2 σ 2 ) . {\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp {\left(-{\frac…

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AAD
σ 2 / π {\textstyle \sigma {\sqrt {2/\pi }}}
CDF
Φ ( x − μ σ ) = 1 2 [ 1 + erf ⁡ ( x − μ σ 2 ) ] {\displaystyle \Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma…
CF
exp ⁡ ( i μ t − σ 2 t 2 / 2 ) {\displaystyle \exp(i\mu t-\sigma ^{2}t^{2}/2)}
Entropy
1 2 log ⁡ ( 2 π e σ 2 ) {\textstyle {\tfrac {1}{2}}\log(2\pi e\sigma ^{2})}
Excess kurtosis
0 {\displaystyle 0}
Fisher information
I ( μ , σ ) = ( 1 / σ 2 0 0 2 / σ 2 ) {\displaystyle {\mathcal {I}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}} I ( μ , σ 2 ) = ( 1 / σ 2 0 0 1…

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Normal distribution

Nodes332
Edges331
Triples251
Avg. degree1.99
Density0.006024
Components1

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Normal distribution

Top relations

related to Generating values from normal distribution · 33
Normal distribution → Again, All, An, Box, Compute, Generate, Hadamard, Hall, Hart, If, If X2, In, Integer, Irwin, Monte-Carlo, Muller, Note, Only, Optional, Some
related to Extensions · 21
Normal distribution → All, As, Brownian, Case, Ck, Complex, Euclidean, Gaussian, Hilbert, Kaniadakis, Matrix, Rectified Gaussian, Rk, Several Gaussian, The, The Kaniadakis, These, This, Tsallis, Uhlenbeck
related to Development · 20
Normal distribution → Although, Chances, De Moivre, Delta, Gauss, If, In, Interval, Logarithm, Moivre, Not, Quantity, Ratio, Some, Starting, Stigler, Term, The Doctrine, Theoria, Using
related to Other properties · 19
Normal distribution → EF, For, Gamma, If, In, Józef Marcinkiewicz, KL, Leibler, Many, Marcinkiewicz, NEF, NEF-QVF, Note, Poisson, Specifically, The, The Fisher, The Hellinger, The Kullback
see also · 16
Normal distribution → Abelian, Bates, Bhattacharyya, Fisher, Fox, Gaussian, Hall, Irwin, Kac, Mathematics, Normally, Psi, The, Tweedie, Wrapped, Wright Psi
related to Naming · 11
Normal distribution → Around, English, Gauss, Gaussian, However, Laplace, Laplace's, Other, Pearson, Peirce, Today
related to Confidence intervals · 9
Normal distribution → Approximate, By Cochran's, In, Inverting, Student's, The, There, These, This
related to Approximate normality · 8
Normal distribution → Approximately, Bose, Einstein, In, Poisson, The, Thermal, When
related to Normality tests · 7
Normal distribution → Diagnostic, H0, Ha, Many, Normality, The, Typically
is a · 6
Normal distribution → member of the family of Tweedie exponential dispersion models.Wrapped normal distribution, only distribution where the mean and variance calculated from a set of independent draws are independent of each other.The normal distribution is a subclass of the elliptical di…, only distribution whose cumulants beyond the first two, only distribution with a finite number, poor model.A normal distribution is sometimes informally called a bell curve, special case of the elliptical distributions

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Important terminology

normal distribution displaystyle textstyle mean variance mu sigma frac standard function random distributions independent distributed sqrt sum right probability left

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Normal distributionAADσ 2 / π {\textstyle \sigma {\sqrt {2/\pi }}}1.00infobox
Normal distributionCDFΦ ( x − μ σ ) = 1 2 [ 1 + erf ⁡ ( x − μ σ 2 ) ] {\displaystyle \Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma…1.00infobox
Normal distributionCFexp ⁡ ( i μ t − σ 2 t 2 / 2 ) {\displaystyle \exp(i\mu t-\sigma ^{2}t^{2}/2)}1.00infobox
Normal distributionEntropy1 2 log ⁡ ( 2 π e σ 2 ) {\textstyle {\tfrac {1}{2}}\log(2\pi e\sigma ^{2})}1.00infobox
Normal distributionExcess kurtosis0 {\displaystyle 0}1.00infobox
Normal distributionFisher informationI ( μ , σ ) = ( 1 / σ 2 0 0 2 / σ 2 ) {\displaystyle {\mathcal {I}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}} I ( μ , σ 2 ) = ( 1 / σ 2 0 0 1…1.00infobox
Normal distributionKullback–Leibler divergence1 2 { ( σ 0 σ 1 ) 2 + ( μ 1 − μ 0 ) 2 σ 1 2 − 1 + ln ⁡ σ 1 2 σ 0 2 } {\displaystyle {1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _…1.00infobox
Normal distributionMADσ 2 erf − 1 ⁡ ( 1 / 2 ) {\displaystyle \sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(1/2)}1.00infobox
Normal distributionMeanμ {\displaystyle \mu }1.00infobox
Normal distributionMedianμ {\displaystyle \mu }1.00infobox
Normal distributionMGFexp ⁡ ( μ t + σ 2 t 2 / 2 ) {\displaystyle \exp(\mu t+\sigma ^{2}t^{2}/2)}1.00infobox
Normal distributionModeμ {\displaystyle \mu }1.00infobox
Normal distributionNotationN ( μ , σ 2 ) {\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}1.00infobox
Normal distributionParametersμ ∈ R {\displaystyle \mu \in \mathbb {R} } = mean (location) σ 2 ∈ R > 0 {\displaystyle \sigma ^{2}\in \mathbb {R} _{>0}} = variance (squared scale)1.00infobox
Normal distributionPDF1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 {\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}1.00infobox
Normal distributionQuantileμ + σ 2 erf − 1 ⁡ ( 2 p − 1 ) {\displaystyle \mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)}1.00infobox
Normal distributionSkewness0 {\displaystyle 0}1.00infobox
Normal distributionSupportx ∈ R {\displaystyle x\in \mathbb {R} }1.00infobox
Normal distributionVarianceσ 2 {\displaystyle \sigma ^{2}}1.00infobox
Normal distributionis apoor model.A normal distribution is sometimes informally called a bell curve0.90text
Normal distributionis aonly distribution whose cumulants beyond the first two0.90text
Normal distributionis aonly distribution where the mean and variance calculated from a set of independent draws are independent of each other.The normal distribution is a subclass of the elliptical di…0.90text
Normal distributionis aonly distribution with a finite number0.90text
Normal distributionis aspecial case of the elliptical distributions0.90text
Normal distributionis amember of the family of Tweedie exponential dispersion models.Wrapped normal distribution0.90text

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