方差

Variance $D (X)$
是衡量随机变量或一组数据离散程度的度量，它描述了数据点与其平均值（期望值）的偏差平方的平均值。
方差提供了数据分布的波动或分散程度的信息。

刻画随机变量取值 $X$ 与数学期望的离散程度

定义：

D (X) = E [(X - E (X))^{2}]

标准差/均方差： $\sqrt{D (X)}$

方差公式：

\begin{array}{r} D (X) = E (X^{2}) - (E (X))^{2} \end{array}

\begin{aligned} D (X) & = E [X^{2} + E (X)^{2} - 2 X E (X)] \\ = E (X^{2}) + E (X)^{2} - 2 E (X) E (X) \\ = E (X^{2}) - E (X)^{2} \end{aligned}

概率分布律 $P {X = x_{i}} = p_{i} i = 1, 2, 3, \dots$

\begin{array}{r} D (X) = \sum_{i = 1}^{\infty} (x_{i} - E (X))^{2} \cdot p_{i} \end{array}

概率密度为 $f (x)$

\begin{array}{r} D (X) = \int_{- \infty}^{+ \infty} {[x_{i} - E (X)]}^{2} f (x) d x \end{array}

\begin{array}{r} D (a X \pm b Y) = a^{2} D (X) + b^{2} D (Y) \pm 2 a b E {[X - E (X)] [Y - E (Y)]} \end{array}

最后一项为协方差
若 $X, Y$ 相互独立，则：
$D (a X \pm b Y) = a^{2} D (X) + b^{2} D (Y)$

\begin{aligned} D (a X \pm b Y) & = E {[a X + b Y - (a E X + b E Y)]}^{2} \\ = E [a (X - E (X)) + b (Y - E (Y))]^{2} \\ = a^{2} E (X - E (X))^{2} + b^{2} E (Y - E (Y))^{2} + 2 a b E [(X - E (X)) (Y - E (Y))] \\ = a^{2} D (X) + b^{2} D (Y) + 2 a b C o v (X, Y) \end{aligned}

随机变量有数学期望 $E (X) = μ$ 方差 $D (X) = σ^{2}$ ，记 $X^{*} = \frac{X - μ}{σ}$

\begin{array}{r} X^{*} = \frac{X - E (X)}{\sqrt{D (X)}} \end{array}

$E (X^{*}) = \frac{1}{σ} [E (X) - μ] = 0$

$D (X^{*}) = \frac{1}{σ^{2}} E [(X - μ)^{2}] = \frac{σ^{2}}{σ^{2}} = 1$

\begin{array}{r} \int_{- \infty}^{+ \infty} x^{2} \frac{1}{\sqrt{2 π}} e^{- \frac{x^{2}}{2}} d x = E (X^{2}) = 1 \end{array}