B Probability distributions

Name	parameters	support	pdf/pmf	mean	variance
Bernoulli	\(p \in [0,1]\)	\(k \in \{0,1\}\)	\(p^k (1 - p)^{1 - k}\) 1.12	\(p\) 7.1	\(p(1-p)\) 7.1
binomial	\(n \in \mathbb{N}\), \(p \in [0,1]\)	\(k \in \{0,1,\dots,n\}\)	\(\binom{n}{k} p^k (1 - p)^{n - k}\) 4.4	\(np\) 7.2	\(np(1-p)\) 7.2
Poisson	\(\lambda > 0\)	\(k \in \mathbb{N}_0\)	\(\frac{\lambda^k e^{-\lambda}}{k!}\) 4.6	\(\lambda\) 7.3	\(\lambda\) 7.3
geometric	\(p \in (0,1]\)	\(k \in \mathbb{N}_0\)	\(p(1-p)^k\) 4.5	\(\frac{1 - p}{p}\) 7.4	\(\frac{1 - p}{p^2}\) 9.3
normal	\(\mu \in \mathbb{R}\), \(\sigma^2 > 0\)	\(x \in \mathbb{R}\)	\(\frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x - \mu)^2}{2 \sigma^2}}\) 4.12	\(\mu\) 7.8	\(\sigma^2\) 7.8
uniform	\(a,b \in \mathbb{R}\), \(a < b\)	\(x \in [a,b]\)	\(\frac{1}{b-a}\) 4.9	\(\frac{a+b}{2}\)	\(\frac{(b-a)^2}{12}\)
beta	\(\alpha,\beta > 0\)	\(x \in [0,1]\)	\(\frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{\text{B}(\alpha, \beta)}\) 4.10	\(\frac{\alpha}{\alpha + \beta}\) 7.6	\(\frac{\alpha \beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}\) 7.6
gamma	\(\alpha,\beta > 0\)	\(x \in (0, \infty)\)	\(\frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1}e^{-\beta x}\) 4.11	\(\frac{\alpha}{\beta}\) 7.5	\(\frac{\alpha}{\beta^2}\) 7.5
exponential	\(\lambda > 0\)	\(x \in [0, \infty)\)	\(\lambda e^{-\lambda x}\) 4.8	\(\frac{1}{\lambda}\) 7.7	\(\frac{1}{\lambda^2}\) 7.7
logistic	\(\mu \in \mathbb{R}\), \(s > 0\)	\(x \in \mathbb{R}\)	\(\frac{e^{-\frac{x - \mu}{s}}}{s(1 + e^{-\frac{x - \mu}{s}})^2}\) 4.13	\(\mu\)	\(\frac{s^2 \pi^2}{3}\)
negative binomial	\(r \in \mathbb{N}\), \(p \in [0,1]\)	\(k \in \mathbb{N}_0\)	\(\binom{k + r - 1}{k}(1-p)^r p^k\) 4.7	\(\frac{rp}{1 - p}\) 9.2	\(\frac{rp}{(1 - p)^2}\) 9.2
multinomial	\(n \in \mathbb{N}\), \(k \in \mathbb{N}\) \(p_i \in [0,1]\), \(\sum p_i = 1\)	\(x_i \in \{0,..., n\}\), \(i \in \{1,...,k\}\), \(\sum{x_i} = n\)	\(\frac{n!}{x_1!x_2!...x_k!} p_1^{x_1} p_2^{x_2}...p_k^{x_k}\) 8.1	\(np_i\)	\(np_i(1-p_i)\)