问题及解答

这里假设随机变量 $\ell_i$ 从属于均匀分布. 即 $\ell_i$ 的概率密度函数为 $p(\ell_i)=\frac{1}{n}$.

计算方差所基于的公式是

\[
\begin{split}
&\frac{1}{n}\sum_{i=1}^{n}(\ell_i-\bar{\ell})^2\\
=\ &\frac{1}{n}\sum_{i=1}^{n}(\ell_i^2-2\ell_i \bar{\ell}+{\bar{\ell}}^2)\\
=\ &\frac{1}{n}\Bigl(\sum_{i=1}^{n}\ell_i^2-2\bar{\ell}\sum_{i=1}^{n}\ell_i+n{\bar{\ell}}^2\Bigr)\\
=\ &\frac{1}{n}\biggl[\sum_{i=1}^{n}\ell_i^2-\frac{2}{n}(\sum_{i=1}^{n}\ell_i)^2+\frac{1}{n}(\sum_{i=1}^{n}\ell_i^2)^2\biggr]\\
=\ &\frac{1}{n}\biggl[\sum_{i=1}^{n}\ell_i^2-\frac{1}{n}(\sum_{i=1}^{n}\ell_i)^2\biggr]\\
=\ &\frac{1}{n}\sum_{i=1}^{n}\ell_i^2-{\bar{\ell}}^2 .
\end{split}
\]

或者

\[
\begin{split}
&\frac{1}{n}\sum_{i=1}^{n}(\ell_i-\bar{\ell})^2\\
=\ &\frac{1}{n}\sum_{i=1}^{n}(\ell_i^2-2\ell_i \bar{\ell}+{\bar{\ell}}^2)\\
=\ &\frac{1}{n}\Bigl(\sum_{i=1}^{n}\ell_i^2-2\bar{\ell}\sum_{i=1}^{n}\ell_i+n{\bar{\ell}}^2\Bigr)\\
=\ &\frac{1}{n}\Bigl(\sum_{i=1}^{n}\ell_i^2-2\bar{\ell}\cdot n\bar{\ell}+n{\bar{\ell}}^2\Big)\\
=\ &\frac{1}{n}\Bigl(\sum_{i=1}^{n}\ell_i^2-n{\bar{\ell}}^2\Big)\\
=\ &\frac{1}{n}\sum_{i=1}^{n}\ell_i^2-{\bar{\ell}}^2.
\end{split}
\]

请描述相应的算法