定义
正态分布的期望和方差
期望
[EX = mu
]
证明
设随机变量 (X sim N(mu, sigma^2)), 求 (EX).
解
[EX = int_{-infty}^{+infty}xf(x)dx = displaystylefrac{1}{sqrt{2pi}sigma}int_{-infty}^{+infty}(sigma t + mu)e^{-frac{t^2}{2} }dt
]
[= displaystylefrac{1}{sqrt{2pi}}int_{-infty}^{+infty}sigma t e^{-frac{t^2}{2}}dt + displaystylefrac{mu}{sqrt{2pi}}int_{-infty}^{+infty}e^{-frac{t^2}{2}}dt = mu,
]
即 (EX = mu).
方差
[DX = sigma^2
]
证明
设随机变量 (X sim N(mu, sigma^2)), 求 (DX).
解
[DX = int_{-infty}^{+infty}(x - mu^2) imes frac{1}{sqrt{2pi}sigma}expleft [ -frac{1}{2}left ( frac{x - mu}{sigma}
ight )^2
ight ]dx = displaystylefrac{sigma ^2}{sqrt{2pi}}int_{-infty}^{+infty} t^2 e^{-frac{t^2}{2}}dt = displaystylefrac{1}{sqrt{2pi}}sigma^2sqrt{2 pi} = sigma^2.
]
重要的性质
一般地, 若 (X_1,...,X_n) 相互独立, 且 (X_i sim N(mu, sigma^2)), 则
[sum_{i = 1}^{n}a_iX_i sim N(sum_{i = 1}^{n}a_imu_i, sum_{i = 1}^{n}a_1^2sigma_i^2).
]