Common distributions

The distributions below use the parameterisation fixed in each card. The second gamma parameter is a rate, and the second inverse-gamma parameter is the rate in \exp(-\beta/x). The longer kernel table, with the problem-solving workflow, is on the formula lookup page. How a sample from one of these laws is summarised before a model is fitted is on Descriptive statistics.

Discrete

Binomial

Distribution	x\sim\operatorname{Bin}(n,p)
Parameters	n\in\mathbb{Z}^{+}, p\in[0,1]
Support	x=0,1,\ldots,n
Probability mass function	p(x)=\binom{n}{x}p^{x}(1-p)^{n-x}
Expectation	\mathbb{E}(x)=np
Variance	\operatorname{Var}(x)=np(1-p)

Article: Binomial distribution →

Geometric

Distribution	x\sim\operatorname{Geom}(p)
Parameters	p\in(0,1]
Support	x=1,2,3,\ldots
Probability mass function	p(x)=p(1-p)^{x-1}
Expectation	\mathbb{E}(x)=1/p
Variance	\operatorname{Var}(x)=(1-p)/p^{2}

Article: Geometric distribution →

Multinomial

Distribution	x\sim\operatorname{Mult}(n,p)
Parameters	n\in\mathbb{Z}^{+}, p\in[0,1]^{K} with \sum_{k=1}^{K}p_k=1
Support	x_k\in\{0,1,\ldots,n\}, \sum_k x_k=n
Probability mass function	p(x)=\frac{n!}{x_1!\cdots x_K!}p_1^{x_1}\cdots p_K^{x_K}
Expectation	\mathbb{E}(x)=np
Variance	\operatorname{Var}(x_k)=np_k(1-p_k)

Article: Multinomial distribution →

Poisson

Distribution	x\sim\operatorname{Poisson}(\theta)
Parameters	\theta>0
Support	x=0,1,2,\ldots
Probability mass function	p(x)=e^{-\theta}\theta^{x}/x!
Expectation	\mathbb{E}(x)=\theta
Variance	\operatorname{Var}(x)=\theta

Article: Poisson distribution →

Continuous

Beta

Distribution	x\sim\operatorname{Beta}(\alpha,\beta)
Parameters	\alpha,\beta>0
Support	0<x<1
Probability density	p(x)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}
Expectation	\mathbb{E}(x)=\alpha/(\alpha+\beta)
Variance	\operatorname{Var}(x)=\alpha\beta/((\alpha+\beta)^{2}(\alpha+\beta+1))

Article: Beta distribution →

Dirichlet

Distribution	x\sim\operatorname{Dirichlet}(\alpha)
Parameters	\alpha\in\mathbb{R}_{+}^{K}, \alpha_0=\sum_{k=1}^{K}\alpha_k
Support	x_k>0, \sum_{k=1}^{K}x_k=1
Probability density	p(x)=\frac{\Gamma(\alpha_0)}{\prod_{k=1}^{K}\Gamma(\alpha_k)}\prod_{k=1}^{K}x_k^{\alpha_k-1}
Expectation	\mathbb{E}(x)=\alpha/\alpha_0
Variance	\operatorname{Var}(x_k)=\alpha_k(\alpha_0-\alpha_k)/(\alpha_0^{2}(\alpha_0+1))

Article: Dirichlet distribution →

Exponential

Distribution	x\sim\operatorname{Exp}(\lambda)
Parameters	\lambda>0
Support	x>0
Probability density	p(x)=\lambda e^{-\lambda x}
Expectation	\mathbb{E}(x)=1/\lambda
Variance	\operatorname{Var}(x)=1/\lambda^{2}

Article: Exponential distribution →

Gamma

Distribution	x\sim\operatorname{Gamma}(\alpha,\beta)
Parameters	\alpha,\beta>0, shape–rate
Support	x>0
Probability density	p(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}
Expectation	\mathbb{E}(x)=\alpha/\beta
Variance	\operatorname{Var}(x)=\alpha/\beta^{2}

Article: Gamma distribution →

Inverse gamma

Distribution	x\sim\operatorname{InvGamma}(\alpha,\beta)
Parameters	\alpha,\beta>0
Support	x>0
Probability density	p(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{-\alpha-1}e^{-\beta/x}
Expectation	\mathbb{E}(x)=\beta/(\alpha-1) for \alpha>1
Variance	\operatorname{Var}(x)=\beta^{2}/((\alpha-1)^{2}(\alpha-2)) for \alpha>2

The mode is \beta/(\alpha+1), which is not the variance.

Article: Inverse-Gamma distribution →

Normal

Distribution	X\sim\operatorname{N}(\mu,\sigma^{2})
Parameters	\mu\in\mathbb{R}, \sigma>0
Support	x\in\mathbb{R}
Probability density	p(x)=\frac{1}{\sqrt{2\pi}\,\sigma}\exp\Bigl(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\Bigr)
Expectation	\mathbb{E}(X)=\mu
Variance	\operatorname{Var}(X)=\sigma^{2}

Article: Normal distribution →