期待値と分散と標準偏差

離散型の場合

$$ P(X=x_k) = f(x_k) \\ \sum_{k=1}^{\infty}f(x_k) = 1 \\ E(X) = \sum_{x}x f(x) \\ V(X) = \sum_{x}(x-\mu)^2 f(x) \\ D(X) = \sqrt{V(X)} \\ $$

連続型の場合

$$ P(X=x_k) = f(x_k) \\ \int_{-\infty}^{\infty} f(x) \space dx = 1 \\ E(X) = \int_{-\infty}^{\infty} xf(x)\space dx \\ V(X) = \int_{-\infty}^{\infty} (x-\mu)^2f(x)\space dx \\ D(X) = \sqrt{V(X)} \\ $$

標準化した場合

$$ Z = \{X-X(X)\} / \sqrt{V(X)} \\ E(Z) = 0, \space V(Z) = 1 $$

$$ E(X) = E(0 \leq X \leq 6) \\ = \int_{0}^{6} xf(x) \space dx \\ = \int_{0}^{6} x\frac{1}{6} \space dx \\ = [\frac{x^2}{12}]_{0}^{6} \
= 3 $$

REFERENCES:

  • https://bellcurve.jp/statistics/course/6712.html