General Error Distribution Wiki
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For other uses, see Bell curve (disambiguation). Normal distribution Probability density function The red curve is the standard normal distribution Cumulative distribution function Notation N ( μ , σ 2 ) {\displaystyle {\mathcal σ 4}(\mu ,\,\sigma ^ σ 3)} Parameters μ ∈ R
Generalized Gaussian Distribution Matlab Code
— mean (location) σ2 > 0 — variance (squared scale) Support x ∈ R generalized normal distribution r PDF 1 2 σ 2 π e − ( x − μ ) 2 2 σ 2 {\displaystyle {\frac σ 0{\sqrt − skewed generalized error distribution 9\pi }}}\,e^{-{\frac {(x-\mu )^ − 8} − 7}}}} CDF 1 2 [ 1 + erf ( x − μ σ 2 ) ] {\displaystyle {\frac − 2 − 1}\left[1+\operatorname − 0 \left({\frac 9{\sigma {\sqrt
Error Distribution Definition
8}}}\right)\right]} Quantile μ + σ 2 erf − 1 ( 2 F − 1 ) {\displaystyle \mu +\sigma {\sqrt 2}\operatorname 1 ^{-1}(2F-1)} Mean μ Median μ Mode μ Variance σ 2 {\displaystyle \sigma ^ − 8\,} Skewness 0 Ex. kurtosis 0 Entropy 1 2 ln ( 2 σ 2 π e ) {\displaystyle {\tfrac − 6 − 5}\ln(2\sigma ^ − 4\pi \,e\,)} MGF exp { μ t
Parametric Generalized Gaussian Density Estimation
+ 1 2 σ 2 t 2 } {\displaystyle \exp\{\mu t+{\frac − 0 σ 9}\sigma ^ σ 8t^ σ 7\}} CF exp { i μ t − 1 2 σ 2 t 2 } {\displaystyle \exp\ σ 2 σ 1}\sigma ^ σ 0t^ μ 9\}} Fisher information ( 1 / σ 2 0 0 1 / ( 2 σ 4 ) ) {\displaystyle {\begin μ 41/\sigma ^ μ 3&0\\0&1/(2\sigma ^ μ 2)\end μ 1}} In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[1][2] The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of random variables is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[3] Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in
is the probability of k occurrences given λ. The function is defined only at integer values of k. The connecting lines are only guides for generalized error distribution r the eye. Cumulative distribution function The horizontal axis is the index k,
Generalized Normal Distribution Python
the number of occurrences. The CDF is discontinuous at the integers of k and flat everywhere else because a gaussian distribution function variable that is Poisson distributed takes on only integer values. Parameters λ > 0 (real) Support k ∈ ℕ pmf λ k e − λ k ! {\displaystyle {\frac {\lambda https://en.wikipedia.org/wiki/Normal_distribution ^ θ 7e^{-\lambda }} θ 6}} CDF Γ ( ⌊ k + 1 ⌋ , λ ) ⌊ k ⌋ ! {\displaystyle {\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}} , or e − λ ∑ i = 0 ⌊ k ⌋ λ i i ! {\displaystyle e^{-\lambda }\sum _ θ 3^{\lfloor k\rfloor }{\frac {\lambda ^ θ 2} θ https://en.wikipedia.org/wiki/Poisson_distribution 1}\ } , or Q ( ⌊ k + 1 ⌋ , λ ) {\displaystyle Q(\lfloor k+1\rfloor ,\lambda )} (for k ≥ 0 {\displaystyle k\geq 0} , where Γ ( x , y ) {\displaystyle \Gamma (x,y)} is the incomplete gamma function, ⌊ k ⌋ {\displaystyle \lfloor k\rfloor } is the floor function, and Q is the regularized gamma function) Mean λ {\displaystyle \lambda } Median ≈ ⌊ λ + 1 / 3 − 0.02 / λ ⌋ {\displaystyle \approx \lfloor \lambda +1/3-0.02/\lambda \rfloor } Mode ⌈ λ ⌉ − 1 , ⌊ λ ⌋ {\displaystyle \lceil \lambda \rceil -1,\lfloor \lambda \rfloor } Variance λ {\displaystyle \lambda } Skewness λ − 1 / 2 {\displaystyle \lambda ^{-1/2}} Ex. kurtosis λ − 1 {\displaystyle \lambda ^{-1}} Entropy λ [ 1 − log ( λ ) ] + e − λ ∑ k = 0 ∞ λ k log ( k ! ) k ! {\displaystyle \lambda [1-\log(\lambda )]+e^{-\lambda }\sum _ − 7^{\infty }{\frac {\lambda ^ − 6\log(k!)} − 5}} (for large λ {\displaystyle \lambda } ) 1 2 log
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index] Re: Re: st: Generalized Error Distribution (GED) in Stata From Stas Kolenikov
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