パレート分布の期待値・分散

確率密度関数

$$f_{X}(x) = \beta \alpha x^{-(\beta+1)}, \; x \in[\alpha, \infty)$$

期待値

$1 < \beta$ に注意．

$$\begin{aligned} \mathbb{E}(X) & = \int_{\alpha}^{\infty} x f_{X}(x) d x \\ & = \int_{\alpha}^{\infty} x \beta \alpha^{\beta} x^{-(\beta+1)} d x \\ & = \int \beta \alpha^{\beta} x^{-(\beta+1)+1} d x \\ & = \int \beta \alpha^{\beta} x^{-(\beta-1)-1} d x \\ & = \int \beta(\beta-1)^{-1}(\beta-1) \alpha^{\beta} \alpha^{\beta-1} \alpha^{-\{\beta-1\}} x^{-\{(\beta-1)+1\}} \\ & = \alpha \beta(\beta-1)^{-1} \\ \end{aligned}$$

分散

$2 < \beta$ に注意．

$$\begin{aligned} \mathbb{E}(X^{2}) & = \int_{\alpha}^{\infty} x^{2} f_{X}(x) d x \\ & = \int_{\alpha}^{\infty} x^{2} \beta \alpha^{\beta} x^{-(\beta+1)} d x \\ & =\int \beta \alpha^{\beta} x^{-(\beta+1)+2} d x \\ & =\int \beta \alpha^{\beta} x^{-\{(\beta-2)+1\}} d x \\ & =\int \beta(\beta-2)^{-1}(\beta-2) \alpha^{\beta} \alpha^{\beta-2} \alpha^{-\{\beta-2\}} x^{-\{(\beta-2)+1\}} \\ & =\alpha^{2} \beta(\beta-2)^{-1} \\ \end{aligned}$$

$$\begin{aligned} \mathbb{V}(X) & = \mathbb{E}(X^{2})-\mathbb{E}(X)^{2} \\ & = \alpha^{2} \beta(\beta-2)^{-1}-\alpha^{2} \beta^{2}(\beta-1)^{-2} \\ & = \alpha^{2} \beta(\beta-2)^{-1}(\beta-1)^{-2}(\beta-1)^{2}-\alpha^{2} \beta(\beta-1)^{-2}(\beta-2)^{-1}(\beta-2) \beta \\ & = \alpha^{2} \beta(\beta-2)^{-1}(\beta-1)^{-2}\left\{(\beta-1)^{2}-(\beta-2) \beta\right\} \\ & = \alpha^{2} \beta(\beta-2)^{-1}(\beta-1)^{-2}\left\{\beta^{2}-2 \beta+1-\beta^{2}+2 \beta\right\} \\ & = \alpha^{2} \beta(\beta-2)^{-1}(\beta-1)^{-2} \end{aligned}$$