Distribution Parameters Probability density function Mean Variance
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
$$\frac{1}{x} \cdot \frac{1}{\sigma \sqrt{2\pi}} \exp \left( - \frac{(\ln x - \mu)^2}{2\sigma^2} \right)$$ $$e^{(\mu + \frac{\sigma^{2}}{2})}$$ $$e^{(2\mu + \sigma^2)}(e^{\sigma^2} - 1)$$