期望与方差的定义

\(X\)为随机变量, \(p(x)\)为它的概率密度或质量函数.

期望

\[\mu = E(X) = \int_{-\infty}^{{+\infty}} x p(x) dx \]

方差

\[\sigma^2 = D(X) = \int_{-\infty}^{{+\infty}} (x - \mu)^2 p(x)dx \\= \int_{-\infty}^{{+\infty}} x^2 p(x) dx + \int_{-\infty}^{{+\infty}} \mu^2 p(x) dx - 2\mu \int_{-\infty}^{{+\infty}} x p(x) = E(X^2) - E^2(X) \]

\(\sigma\)为标准差.