定义
期望
[EX = frac{a + b}{2}.
]
证明
[EX = int_{-infty }^{+infty }xf(x)dx = int_{a}^{b}xfrac{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^2frac{1}{b - a}dx = displaystylefrac{b^{3} - a^{3}}{3(b - a)} = displaystylefrac{b^{2} + ab + a^{2}}{3},
]
所以，
[DX = EX^2 - (EX)^2 = displaystylefrac{b^{2} + ab + a^{2}}{3} - (displaystylefrac{a + b}{2})^2 = displaystylefrac{(b - a)^2}{12}.
]