variance 方差：到均值距离的平方的平均值 the expected value of the squared devitation from the mean of $X,\mu = \mathrm E[X]$：
$$ \mathrm{Var}(X)=\mathrm{E}\big[(X-\mu)^{2}\big].\\=\mathrm{Cov}(X,X)=\sigma^2=\mathrm V(X) $$
$$ \begin{aligned}\operatorname{Var}(X)& =\operatorname{E}\big[(X-\operatorname{E}[X])^{2}\big] \\&=\operatorname{E}\big[X^{2}-2X\operatorname{E}[X]+\operatorname{E}[X]^{2}\big] \\&=\operatorname{E}\big[X^{2}\big]-2\operatorname{E}[X]\operatorname{E}[X]+\operatorname{E}[X]^{2} \\&=\mathrm{E}\big[X^{2}\big]-\mathrm{E}[X]^{2}\end{aligned} $$
Expectation 均值
<aside> 🤡 [language of measure theory] If $X$ is a real-valued random variable defiend on a probability space $(\Omega,\Sigma,\mathrm P)$, then the expected value of $X$, denoted by $\mathrm E[X]$, is defined as Lebesgue integral: $\mathrm E[X]=\int_\Omega X d\mathrm P$
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Covariance 协方差
Probability Space 概率空间