$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$\mathrm{Likelihood}\left('\mathrm{{\rm B}}'\left(a\,b\right)\,A\,\mathrm{samplesize}=1\right)$

$\left\{\begin{array}{cc}0& A_1<0\\ \frac{A_1^{a-1}{\left(1-A_1\right)}^{b-1}}{\mathrm{{\rm B}}\left(a\&comma;b\right)}& A_1<1\\ 0& 1\le A_1\end{array}\right.$

$\mathrm{LogLikelihood}\left('\mathrm{{\rm B}}'\left(a\&comma;b\right)\&comma;A\&comma;\mathrm{samplesize}=1\right)$

$\ln \left(\frac{A_1^{a-1}{\left(1-A_1\right)}^{b-1}}{\mathrm{{\rm B}}\left(a\&comma;b\right)}\right)$

$\mathrm{Score}\left('\mathrm{{\rm B}}'\left(a\&comma;b\right)\&comma;A\&comma;\mathrm{samplesize}=1\right)$

$\left[\begin{array}{c}\ln \left(A_1\right)-\mathrm{\Psi }\left(a\right)+\mathrm{\Psi }\left(a+b\right)\\ \ln \left(1-A_1\right)-\mathrm{\Psi }\left(b\right)+\mathrm{\Psi }\left(a+b\right)\end{array}\right]$

$S≔\mathrm{Sample}\left(\mathrm{Binomial}\left(10\&comma;0.4\right)\&comma;1000\right)$

$\mathrm{MaximumLikelihoodEstimate}\left(\mathrm{Binomial}\left(10\&comma;\mathrm{\theta }\right)\&comma;S\&comma;\mathrm{bounds}=0..1\right)$

$0.393299998240230$

$\mathrm{Information}\left(\mathrm{Normal}\left(\mathrm{\mu }\&comma;\mathrm{\sigma }\right)\&comma;A\&comma;\mathrm{samplesize}=1\&comma;\mathrm{param}=\mathrm{\mu }\right)$

$-\frac{1}{{\mathrm{\sigma }}^2}$

$\mathrm{FisherInformation}\left(\mathrm{Normal}\left(\mathrm{\mu }\&comma;\mathrm{\sigma }\right)\&comma;1\&comma;\mathrm{\mu }\right)$

$\frac{1}{{\mathrm{\sigma }}^2}$

$\mathrm{LikelihoodRatioStatistic}\left(\mathrm{Normal}\left(\mathrm{\mu }\&comma;5\right)\&comma;A\&comma;\mathrm{samplesize}=1\right)$

$\frac{1}{25}{A}_1^2-\frac{2}{25}{A}_1\mathrm{\mu }+\frac{1}{25}{\mathrm{\mu }}^2$