上分位点

Upper Quantile

对于给定的概率 $α \in (0, 1)$ ，随机变量 $X$
满足 $P {X > x_{α}} = α$ 的实数指标 $x_{α}$ 为 $X$ 的概率分布的上分位数

\begin{array}{r} P {X > x_{α}} = α \end{array}

\begin{aligned} X \sim N (0, 1) \\ P {X > x_{α}} & = α \\ 1 - P {X \leq x_{α}} & = 1 - Φ (x_{α}) \\ Φ (x_{α}) & = 1 - α \end{aligned}

\begin{aligned} P {| X | \leq λ} & = 2 Φ (λ) - 1 = 1 - α \\ Φ (λ) & = 1 - \frac{α}{2} \\ λ & = x_{\frac{α}{2}} \end{aligned}

$P {χ^{2} > χ_{α}^{2} (n)} = α$
n 充分大时， $χ_{α}^{2} (n) \approx \frac{1}{2} (z_{α} + \sqrt{2 n - 1})^{2}$
$P {χ^{2} < λ} = α λ = χ_{1 - λ}^{2}$