| Bernoulli |
B_r(q) |
q^x(1-q)^{1-x} |
x=0,1 |
0\le q \le 1 |
q |
0 or 1 |
0 or 1 |
q(1-q) |
scipy.stats.bernoulli.pmf |
| Poisson |
Po(\lambda) |
\Large\frac{\lambda^x e^{-\lambda}}{x!} |
x=0,1,2,... |
\lambda >0 |
\lambda |
- |
\lceil \lambda\rceil-1,\lfloor\lambda\rfloor |
\lambda |
scipy.stats.poisson.pmf |
| Uniform |
U(a,b) |
\large\frac{1}{b-a} |
a\le x\le b |
-\infty |
\large\frac{b-a}{2} |
\large\frac{b-a}{2} |
[a,b] |
\large\frac{(b-a)^2}{12} |
scipy.stats.uniform.pdf |
| beta |
Be(\alpha,\beta) |
\large\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} |
0\le x\le 1 |
\alpha>0,\beta > 0 |
\large\frac{\alpha}{\alpha + \beta} |
- |
\large\frac{\alpha - 1}{\alpha + \beta - 2} |
\large\frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)} |
scipy.stats.beta.pdf |
| regular |
N(\mu,\sigma^2) |
{\large\frac{1}{\sqrt{2\pi\sigma^2}}}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right] |
x\in \mathbb{R} |
\mu\in \mathbb{R},\sigma^2 > 0 |
\mu |
\mu |
\mu |
\sigma^2 |
scipy.stats.norm.pdf |
| t |
T(\nu,\mu,\sigma^2) |
\frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\sqrt{}\nu\pi\sigma^2}\left[1+\frac{(x-\mu)^2}{\nu\sigma^2}\right]^{-\frac{\nu+1}{2}} |
x\in \mathbb{R} |
\nu>0,\mu\in\mathbb{R},\sigma^2>0 |
\mu |
\mu |
\mu |
{\large\frac{\nu}{\nu-2}}\sigma^2 |
scipy.stats.t.pdf |
| Cauchy |
Ca(\mu,\sigma) |
{\large\frac{1}{\pi\sigma} }\left[ 1+ {\large\left(\frac{x-\mu}{\sigma}\right)}^2 \right]^{-1} |
x\in \mathbb{R} |
\mu\in \mathbb{R},\sigma > 0 |
n.a. |
\mu |
\mu |
n.a. |
scipy.stats.cauchy.pdf |
| Laplace |
La(\mu,\sigma) |
{\large\frac{1}{2\sigma}} \exp \left[ -{\large\frac{|x-\mu|}{\sigma}} \right] |
x\in \mathbb{R} |
\mu\in \mathbb{R},\sigma > 0 |
\mu |
\mu |
\mu |
2\sigma^2 |
scipy.stats.laplace.pdf |
| gamma |
Ga(\alpha,\beta) |
{\large\frac{\beta^{\alpha}}{\Gamma(\alpha)}}x^{\alpha-1}e^{-\beta x} |
x>0 |
\alpha >0,\beta>0 |
{\large\frac{\alpha}{\beta}} |
- |
{\large\frac{\alpha-1}{\beta}} |
{\large\frac{\alpha}{\beta^2}} |
scipy.stats.gamma.pdf |
| Inverse gamma |
Ga^{-1}(\alpha,\beta) |
{\large\frac{\beta^{\alpha}}{\Gamma(\alpha)}}x^{-(\alpha+1)}e^{-\frac{\beta}{x}} |
x>0 |
\alpha >0,\beta>0 |
{\large\frac{\beta}{\alpha-1}} |
- |
{\large\frac{\beta}{\alpha+1}} |
{\large\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}} |
scipy.stats.invgamma.pdf |
| Kai squared |
X^2(\nu) |
\frac{\left(\frac{1}{2}\right)^{\frac{\nu}{2}} }{\Gamma\left(\frac{\nu}{2}\right)}x^{\frac{\nu}{2}-1}e^{-\frac{x}{2}} |
x>0 |
\nu>0 |
\nu |
- |
\nu-2 |
2\nu |
scipy.stats.chi2.pdf |
| index |
\epsilon_{xp}(\lambda) |
\lambda e^{-\lambda x} |
x>0 |
\lambda>0 |
{\large\frac{1}{\lambda}} |
{\large\frac{\log{2}}{\lambda}} |
0 |
{\large\frac{1}{\lambda^2}} |
scipy.stats.expon.pdf |
| Multivariate normal |
N_m({\bf\mu},{\bf\Sigma}) |
{\large\frac{1}{(2\pi)^{\frac{m}{2}}\sqrt{|{\bf\Sigma}|}}} \exp \left\[-\frac{1}{2}({\bf x}-{\bf \mu})^{\top}{\bf\Sigma}^{-1}({\bf x}-{\bf \mu})\right\] |
{\bf x} \in\mathbb{R}^m |
{\bf\mu} \in\mathbb{R}^m,|{\bf\Sigma}|>0 |
{\bf\mu} |
- |
{\bf\mu} |
{\bf\Sigma} |
scipy.stats.multivariate_normal.pdf |
| Multivariate t |
T_m(\nu,{\bf\mu},{\bf\Sigma}) |
\frac{\Gamma\left({\frac{\nu+m}{2}}\right)}{\Gamma\left({\frac{\nu}{2}}\right) (\pi\nu)^{\frac{m}{2}} |{\bf\Sigma}|^{\frac{1}{2}}} \left[ 1 + \frac{1}{\nu} (\textbf{x} - {\bf \mu})^{T} {\bf \Sigma}^{-1}(\textbf{x} - {\bf \mu})\right]^{\frac{-(\nu+m)}{2}} |
{\bf x} \in \mathbb{R}^m |
\nu>0,{\bf\mu} \in\mathbb{R}^m,|{\bf\Sigma}|>0 |
{\bf\mu} |
- |
{\bf\mu} |
{\large\frac{\nu}{\nu-2}}{\bf\Sigma} |
(investigating..) |