We define a new quantity
$$ E_L(\boldsymbol{R};\boldsymbol{\alpha})=\frac{1}{\psi_T(\boldsymbol{R};\boldsymbol{\alpha})}H\psi_T(\boldsymbol{R};\boldsymbol{\alpha}), \tag{1} $$called the local energy, which, together with our trial PDF yields a new expression (and which look simlar to the the expressions for moments in statistics)
$$ \overline{E}[\boldsymbol{\alpha}]=\int P(\boldsymbol{R})E_L(\boldsymbol{R};\boldsymbol{\alpha}) d\boldsymbol{R}\approx \frac{1}{N}\sum_{i=1}^NE_L(\boldsymbol{R_i};\boldsymbol{\alpha}) \tag{2} $$with \( N \) being the number of Monte Carlo samples. The expression on the right hand side follows from Bernoulli's law of large numbers, which states that the sample mean, in the limit \( N\rightarrow \infty \) approaches the true mean