>Michi zh: /* Normal distribution */

2020-01-16T22:14:35Z

Normal distribution

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{{Testcases notice}}

== [[Normal distribution]] ==
{{Test case|_format=columns
| name = Normal Distribution
| type = density
| pdf_image = [[File:Normal Distribution PDF.svg|340px|Probability density function for the normal distribution]] The red curve is the ''standard normal distribution''
| cdf_image = [[File:Normal Distribution CDF.svg|340px|Cumulative distribution function for the normal distribution]]
| notation = <math>\mathcal{N}(\mu,\sigma^2)</math>
| parameters = <math>\mu\in\R</math> = mean ([[location parameter|location]]) <math>\sigma^2>0</math> = variance (squared [[scale parameter|scale]])
| support = <math>x\in\R</math>
| pdf = <math>\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{2 \sigma^2}}</math>
| cdf = <math>\frac{1}{2}\left[1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right] </math>
| quantile = <math>\mu+\sigma\sqrt{2} \operatorname{erf}^{-1}(2p-1)</math>
| mean = <math>\mu</math>
| median = <math>\mu</math>
| mode = <math>\mu</math>
| variance = <math>\sigma^2</math>
| mad = <math>\sqrt{2/\pi}\sigma</math>
| skewness = <math>0</math>
| kurtosis = <math>0</math> 
| entropy = <math>\frac{1}{2} \log(2\pi e\sigma^2)</math>
| mgf = <math>\exp(\mu t + \sigma^2t^2/2)</math>
| char = <math>\exp(i\mu t - \sigma^2 t^2/2)</math>
| fisher = <math>\mathcal{I}(\mu,\sigma) =\begin {pmatrix} 1/\sigma^2 & 0 \\ 0 & 2/\sigma^2\end{pmatrix}</math> 
<math>\mathcal{I}(\mu,\sigma^2) =\begin {pmatrix} 1/\sigma^2 & 0 \\ 0 & 1/(2\sigma^4)\end{pmatrix}</math>
| KLDiv = <math>D_\text{KL}(\mathcal{N}_0 \| \mathcal{N}_1) = { 1 \over 2 } \{ (\sigma_0/\sigma_1)^2 + \frac{(\mu_1 - \mu_0)^2}{\sigma_1^2} - 1 + 2 \ln {\sigma_1 \over \sigma_0} \}</math>
}}

== [[Binomial distribution]] ==
{{Test case|_format=columns
| name = Binomial distribution
| type = mass
| pdf_image = [[File:Binomial distribution pmf.svg|300px|Probability mass function for the binomial distribution]]
| cdf_image = [[File:Binomial distribution cdf.svg|300px|Cumulative distribution function for the binomial distribution]]
| notation = <math>B(n,p)</math>
| parameters = <math>n \in \{0, 1, 2, \ldots\}</math> – number of trials <math>p \in [0,1]</math> – success probability for each trial
| support = <math>k \in \{0, 1, \ldots, n\}</math> – number of successes
| pdf = <math>\binom{n}{k} p^k (1-p)^{n-k}</math>
| cdf = <math>I_{1-p}(n - k, 1 + k)</math>
| mean = <math>np</math>
| median = <math>\lfloor np \rfloor</math> or <math>\lceil np \rceil</math>
| mode = <math>\lfloor (n + 1)p \rfloor</math> or <math>\lceil (n + 1)p \rceil - 1</math>
| variance = <math>np(1 - p)</math>
| skewness = <math>\frac{1-2p}{\sqrt{np(1-p)}}</math>
| kurtosis = <math>\frac{1-6p(1-p)}{np(1-p)}</math>
| entropy = <math>\frac{1}{2} \log_2 \left( 2\pi enp(1-p) \right) + O \left( \frac{1}{n} \right)</math> in [[Shannon (unit)|shannons]]. For [[nat (unit)|nats]], use the natural log in the log.
| mgf = <math>(1-p + pe^t)^n</math>
| char = <math>(1-p + pe^{it})^n</math>
| pgf = <math>G(z) = [(1-p) + pz]^n</math>
| fisher = <math> g_n(p) = \frac{n}{p(1-p)} </math> (for fixed <math>n</math>)
}}

== [[Geometric distribution]] ==
{{Test case|_format=columns
| name = Geometric
| type = mass
| pdf_image = [[File:geometric pmf.svg|450px]]
| cdf_image = [[File:geometric cdf.svg|450px]]
| parameters = <math>0 success probability ([[real number|real]])
| support = ''k'' trials where <math>k \in \{1,2,3,\dots\}</math>
| pdf = <math>(1 - p)^{k-1}p</math>
| cdf = <math>1-(1 - p)^k</math>
| mean = <math>\frac{1}{p}</math>
| median = <math>\left\lceil \frac{-1}{\log_2(1-p)} \right\rceil</math> 
(not unique if <math>-1/\log_2(1-p)</math> is an integer)
| mode = <math>1</math>
| variance = <math>\frac{1-p}{p^2}</math>
| skewness = <math>\frac{2-p}{\sqrt{1-p}}</math>
| kurtosis = <math>6+\frac{p^2}{1-p}</math>
| entropy = <math>\tfrac{-(1-p)\log_2 (1-p) - p \log_2 p}{p}</math>
| mgf = <math>\frac{pe^t}{1-(1-p) e^t},</math> for <math>t<-\ln(1-p)</math>
| char = <math>\frac{pe^{it}}{1-(1-p)e^{it}}</math>
| parameters2 = <math>0 success probability ([[real number|real]])
| support2 = ''k'' failures where <math>k \in \{0,1,2,3,\dots\}</math>
| pdf2 = <math>(1 - p)^k p</math>
| cdf2 = <math>1-(1 - p)^{k+1}</math>
| mean2 = <math>\frac{1-p}{p}</math>
| median2 = <math>\left\lceil \frac{-1}{\log_2(1-p)} \right\rceil - 1</math> 
(not unique if <math>-1/\log_2(1-p)</math> is an integer)
| mode2 = <math>0</math>
| variance2 = <math>\frac{1-p}{p^2}</math>
| skewness2 = <math>\frac{2-p}{\sqrt{1-p}}</math>
| kurtosis2 = <math>6+\frac{p^2}{1-p}</math>
| entropy2 = <math>\tfrac{-(1-p)\log_2 (1-p) - p \log_2 p}{p}</math>
| mgf2 = <math>\frac{p}{1-(1-p)e^t}</math>
| char2 = <math>\frac{p}{1-(1-p)e^{it}}</math>
}}

== [[Gamma distribution]] ==
{{Test case|_format=columns
| name =Gamma
| type =density
| pdf_image =[[File:Gamma distribution pdf.svg|325px|Probability density plots of gamma distributions]]
| cdf_image =[[File:Gamma distribution cdf.svg|325px|Cumulative distribution plots of gamma distributions]]
| parameters =
* ''k'' > 0 [[shape parameter|shape]]
* ''θ'' > 0 [[scale parameter|scale]]
| support =<math>x \in (0, \infty)</math>
| pdf =<math>\frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta}}</math>
| cdf =<math>\frac{1}{\Gamma(k)} \gamma\left(k, \frac{x}{\theta}\right)</math>
| mean =<math>\operatorname{E}[X] = k \theta </math>
| median =No simple closed form
| mode =<math>(k - 1)\theta \text{ for } k \geq 1</math>
| variance =<math>\operatorname{Var}(X) = k \theta^2</math>
| skewness =<math>\frac{2}{\sqrt{k}}</math>
| kurtosis =<math>\frac{6}{k}</math>
| entropy =<math>\begin{align}
k &+ \ln\theta + \ln\Gamma(k)\\
&+ (1 - k)\psi(k)
\end{align}</math>
| mgf =<math>(1 - \theta t)^{-k} \text{ for } t < \frac{1}{\theta}</math>
| char =<math>(1 - \theta it)^{-k}</math>
| parameters2 =
* ''α'' > 0 [[shape parameter|shape]]
* ''β'' > 0 [[rate parameter|rate]]
| support2 =<math>x \in (0, \infty)</math>
| pdf2 =<math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x }</math>
| cdf2 =<math>\frac{1}{\Gamma(\alpha)} \gamma(\alpha, \beta x)</math>
| mean2 =<math>\operatorname{E}[X] = \frac{\alpha}{\beta}</math>
| median2 =No simple closed form
| mode2 =<math>\frac{\alpha - 1}{\beta} \text{ for } \alpha \geq 1</math>
| variance2 =<math>\operatorname{Var}(X) = \frac{\alpha}{\beta^2}</math>
| skewness2 =<math>\frac{2}{\sqrt{\alpha}}</math>
| kurtosis2 =<math>\frac{6}{\alpha}</math>
| entropy2 =<math>\begin{align}
\alpha &- \ln \beta + \ln\Gamma(\alpha)\\
&+ (1 - \alpha)\psi(\alpha)
\end{align}</math>
| mgf2 =<math>\left(1 - \frac{t}{\beta}\right)^{-\alpha} \text{ for } t < \beta</math>
| char2 =<math>\left(1 - \frac{it}{\beta}\right)^{-\alpha}</math>
}}

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>Michi zh: /* Normal distribution */