[Math Note] #Math

calculation.md

Square of sumation $$ \left(\sum_{j=0}^{n-1} Z_{j}\right)^{2}=\sum_{j=0}^{n-1} Z_{j}^{2}+\sum_{j \neq i} Z_{i} Z_{j}=\sum_{j=0}^{n-1} Z_{j}^{2}+2 \sum_{j<i} Z_{i} Z_{j} $$ A detailed deduction may be $$ \begin{aligned} \left(\sum_{j=0}^{n-1} Z_{j}\right)^{2} &=\left(\sum_{i=0}^{n-1} Z_{i}\right)\left(\sum_{j=0}^{n-1} Z_{j}\right) \ &=\sum_{j=0}^{n-1} Z_{j}\left(Z_{j}+\sum_{j \neq i} Z_{i}\right) \ &=\sum_{j=0}^{n-1} Z_{j}^{2}+\sum_{j \neq i} Z_{i} Z_{j} \end{aligned} $$

Moment.md

Moment for Gaussian distributed variable $X \sim \mathcal{N}\left(0, \sigma^{2}\right)$ $$E\left[X^{2 n}\right]=(2 n-1) ! ! \sigma^{2 n}$$