正态分布

$$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} \exp \left\{-\frac{1}{2 \sigma^{2}}(x-\mu)^{2}\right\}, \quad x \in \mathbb{R} $$
$$ \begin{array}{lll} \text{Distribution} & \text{Mean} & \text{Variance} \\ \hline \text{Point mass at } p & a & 0 \\ \text{Bernoulli}(p) & p & p(1-p) \\ \text{Binomial}(n, p) & np & np(1-p) \\ \text{Geometric}(p) & 1/p & (1 - p)/p^2 \\ \text{Poisson}(\lambda) & \lambda & \lambda \\ \text{Uniform}(a, b) & (a + b) / 2 & (b - a)^2 / 12 \\ \text{Normal}(\mu, \sigma^2) & \mu & \sigma^2 \\ \text{Exponential}(\beta) & \beta & \beta^2 \\ \text{Gamma}(\alpha, \beta) & \alpha \beta & \alpha \beta^2 \\ \text{Beta}(\alpha, \beta) & \alpha / (\alpha + \beta) & \alpha \beta / ((\alpha + \beta)^2 (\alpha + \beta + 1)) \\ t_\nu & 0 \text{ (if } \nu > 1 \text{)} & \nu / (\nu - 2) \text{ (if } \nu > 2 \text{)} \\ \chi^2_p & p & 2p \\ \text{Multinomial}(n, p) & np & \text{see below} \\ \text{Multivariate Nornal}(\mu, \Sigma) & \mu & \Sigma \\ \end{array} $$
$$ \begin{equation} \begin{array}{ll}\text { Distribution } & \mathrm{MGF} \psi(t) \\ \hline \operatorname{Bernoulli}(p) & p e^{t}+(1-p) \\ \operatorname{Binomial}(n, p) & \left(p e^{t}+(1-p)\right)^{n} \\ \operatorname{Poisson}(\lambda) & e^{\lambda\left(e^{t}-1\right)} \\ \operatorname{Normal}(\mu, \sigma) & \exp \left\{\mu t+\dfrac{\sigma^{2} t^{2}}{2}\right\} \\ \operatorname{Gamma}(\alpha, \beta) & \left(\dfrac{1}{1-\beta t}\right)^{\alpha} \text { for } t<1 / \beta\end{array} \end{equation} $$
Dist Mean Var $\mathbb{E}(X^2)$ PDF/PMF CDF $M_X(t)$ $M_X'(t)$ $M_X''(t)$
$\text{Bernoulli}(p) $ $p$ $p(1-p)$ $p^x (1-p)^{1-x}$ $pe^t + (q-p)$
$\text{Binomial}(n, p) $ $np$ $np(1-p)$ ${n \choose k}p^{k}(1-p)^{n-k}\!$ $(p e^{t}+(1-p))^{n}$
$\text{Geometric}(p) $ ${\frac {1}{p}}\!$ ${\frac {1-p}{p^{2}}}\!$ $(1-p)^{{k-1}}\,p\!$ $1-(1-p)^{k}\!$ ${\frac {pe^{t}}{1-(1-p)e^{t}}}\!$
$\text{Poisson}(\lambda) $ $\lambda $ $\lambda $ $\dfrac{\lambda^x e^{-\lambda}}{x!}$ $\exp(\lambda (e^{{t}}-1))$
$\text{Uniform}(a, b) $ $\dfrac{a+b}{2}$ $\dfrac{(b-a)^{2}}{12}$ $\dfrac{1}{b-a}$ $\dfrac{x-a}{b-a}$ ${\frac {e^{{tb}}-e^{{ta}}}{t(b-a)}}\,\!$
$\text{Normal}(\mu, \sigma^2)$ $\mu $ $\sigma ^{2}$ $\frac1{\sigma\sqrt{2\pi}}\exp\{-\frac{(x- \mu)^2}{2\sigma^2}\}$ $\exp\{\mu t+\dfrac{\sigma^{2}t^{2}}{2}\}$
$\text{Exponential}(\beta) $ $\beta $ $\beta ^{2}$ $\lambda e ^{-\lambda x}$ $1 - e^{-\lambda x}$
$\text{Gamma}(\alpha, \beta) $
$\text{Beta}(\alpha, \beta) $

几何分布:如果每次试验的成功概率是 p,那么 k 次试验中,第 k 次才得到成功的概率是

指数分布:$\lambda$ 单位时间发生次数,比如每天发生多少次。 $\beta = \lambda ^{-1}$: 表示发生一次所经历的周期。

伽马函数

Z 函数

EAT = 150 = p * 100 + (1-p)*280 180p =130