均匀分布
定义
期望
\[EX = \frac{a + b}{2}.
\]
证明
\[EX = \int_{-\infty }^{+\infty }xf(x)dx = \int_{a}^{b}x\frac{1}{b - a} = \frac{1}{b - a}\cdot \frac{b^2 - a^2}{2} = \frac{a + b}{2}.
\]
方差
\[DX = \frac{(b - a)^2}{12}.
\]
证明
\[EX^{2} = \int_{-\infty }^{+\infty }g(x)f(x)dx = \int_{a}^{b}x^2\frac{1}{b - a}dx = \displaystyle\frac{b^{3} - a^{3}}{3(b - a)} = \displaystyle\frac{b^{2} + ab + a^{2}}{3},
\]
所以，
\[DX = EX^2 - (EX)^2 = \displaystyle\frac{b^{2} + ab + a^{2}}{3} - (\displaystyle\frac{a + b}{2})^2 = \displaystyle\frac{(b - a)^2}{12}.
\]
