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概率论与统计学笔记

Summary

极为重要的数学工具,这辈子会经常用到。

主要参考:

  • All of statistics
  • Introduction to Probability

Contents

Cheatsheet
cheatsheet
2021
December 20
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}\!