正态（高斯）分布

$$ X \sim \mathcal{N}(\mu,,\sigma^{2}) $$

$$ p(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}, -\infty < x < \infty $$

$$ F(x) = \frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{x}e^{-\frac{(t-\mu)^2}{2\sigma^2}}dt $$

如果RV $X \sim \mathcal{N}(\mu,,\sigma^{2})$, 则可以化为标准正态分布$U = \frac{X-\mu}{\sigma} \sim \mathcal{N}(0, 1)$

$E[X] = \mu$

$Var[X] = \sigma^2$

• 若$X\sim \mathcal{N}(\mu, \sigma^2)$, $Y=aX+b \sim \mathcal{N}(a\mu+b, a^2\sigma^2)$

• 若$X\sim \mathcal{N}(\mu_1, \sigma_1^2)$, $Y\sim \mathcal{N}(\mu_2, \sigma_2^2)$,

  $Z=X+Y\sim \mathcal{N}(\mu_1+\mu_2, \sigma_1^2+\sigma_2^2)$

均匀分布

$X\sim \mathcal{U}(a,b)$ $$ \begin{equation} p(x) = \begin{cases} \frac{1}{b-a}, &a < x < b \newline 0, &其他 \end{cases} \end{equation} $$

$$ \begin{equation} F(x) = \begin{cases} 0, &a < x \newline \frac{x-a}{b-a}, &a\leq x < b \newline 1, &x \geq b \end{cases} \end{equation} $$

$E[X] = \frac{a+b}{2}$

$Var[X] = \frac{(b-a)^2}{12}$

指数分布

$X \sim Exp(\lambda)$, 其中$\lambda>0$ $$ \begin{equation} p(x) = \begin{cases} \lambda e^{-\lambda x}, &x \geq0 \newline 0, &x<0 \end{cases} \end{equation} $$

$$ \begin{equation} F(x) = \begin{cases} 1 - e^{-\lambda x}, &x\geq0 \newline 0, & x < 0 \end{cases} \end{equation} $$

$$ \begin{equation} \begin{aligned} E[X] &= \frac{1}{\lambda}\newline E[X^2] &= \frac{2}{\lambda^2}\newline Var[X] &= \frac{1}{\lambda^2} \end{aligned} \end{equation} $$ $P(x>s) = e^{-\lambda s}$

无记忆性：$s > 0, t > 0$

$P(X > s + t | X > s) = P(X > t)$