Statistic

Maximum Likelihood

Bayesian Expectation Values

Symbol

Value

Symbol

Value

Limit $n\to \infty$

location m

$\hat{m}$

$\bar{Y}$

$\bar{m}={\langle m\rangle |}_{\left(2\right)}^{\left(n\right)}$

$\bar{Y}$

scale s

$\hat{s}$

$S\equiv \sqrt{\overline{Y^2}-{\bar{Y}}^2}$

$\bar{s}={\langle s\rangle |}_{\left(2\right)}^{\left(n\right)}$

${\left(\frac{n}{2}\right)}^{\frac{1}{2}}\frac{\Gamma \left(\left(n-1\right)/2\right)}{\Gamma \left(n/2\right)}S$

S

$\mathrm{var}\left(m\right)$

${\hat{\sigma }}_m^2$

$S^2/n$

$\Delta m^2={\langle {\left(m-\bar{m}\right)}^2\rangle |}_{\left(2\right)}^{\left(n\right)}$

$\frac{S^2}{n-2}$

$S^2/n$

$\mathrm{var}\left(s\right)$

${\hat{\sigma }}_s^2$

$S^2/2n$

$\Delta s^2={\langle {\left(s-\bar{s}\right)}^2\rangle |}_{\left(2\right)}^{\left(n\right)}$

$\left[\frac{n}{n-2}-\frac{n\Gamma {\left(\left(n-1\right)/2\right)}^2}{2\Gamma {\left(n/2\right)}^2}\right]S^2$

$S^2/2n$

Sample Statistics:

$\bar{Y}=\frac{1}{n}{\displaystyle \underset{k=1}{\overset{n}{\sum }}\ln \left(z_k\right)}$

$\overline{Y^2}=\frac{1}{n}{\displaystyle \underset{k=1}{\overset{n}{\sum }}\ln {\left(z_k\right)}^2}$