| Normal |
\(\mu\) — location
\(\sigma > 0\) — scale |
\(\frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}\) |
\(\mu\) |
\(\sigma^2\) |
| Logistic |
\(\mu\) — location
\(s > 0\) — scale |
\(\frac{e^{-\frac{x-\mu}{s}}}{s \left( 1 + e^{-\frac{x-\mu}{s}} \right)^2} = \frac{1}{4s} \text{ sech}^2 \left(\frac{x - \mu}{2s} \right)\) |
\(\mu\) |
\(\frac{s^{2}\pi^{2}}{3}\) |
| Gamma |
\(\alpha > 0\) — shape
\(\beta > 0\) — rate |
\(\frac{\beta^{\alpha} x^{\alpha - 1} e^{-\beta x}}{\Gamma (\alpha)}\) |
\(\frac{\alpha}{\beta}\) |
\(\frac{\alpha}{\beta^{2}}\) |
| Inverse gauss |
\(\mu > 0\) — location
\(\lambda > 0\) — shape |
\(\left\lbrack \frac{\lambda}{2\pi x^3} \right\rbrack^{1/2} \exp \left\{ \frac{-\lambda (x - \mu)^2}{2\mu^2 x} \right\}\) |
\(\mu\) |
\(\frac{\mu^{3}}{\lambda}\) |
| Lognormal |
\(\mu\) — scale
\(\sigma > 0\) — shape |
\(\frac{1}{x} \cdot \frac{1}{\sigma \sqrt{2\pi}} \exp \left( - \frac{(\ln x - \mu)^2}{2\sigma^2} \right)\) |
\(e^{(\mu + \frac{\sigma^{2}}{2})}\) |
\(e^{(2\mu + \sigma^2)}(e^{\sigma^2} - 1)\) |