1. 基础知识
1.1 Population Mean
\[\mu = \frac{\sum\limits_{i=1}^N{x_i}}{N} = \frac{x_1 + x_2 + … + x_N}{N}\]
1.2 Sample Mean
\[\bar{X} = \frac{\sum\limits_{i=1}^n{x_i}}{n} = \frac{x_1 + x_2 + … + x_n}{n}\]
1.3 Dispersion
\[\sigma^2 = variance\]
\[\sigma ^2 = \frac{\sum\limits_{i=1}^N{(x_i - \mu)}}{N}\]
\[\begin{align} \sigma ^2 &= \frac{\sum\limits_{i=1}^N{(x_i - \mu)}}{N}\\ &= \frac{\sum\limits_{i=1}^N{({x_i}^2 - 2x_i\mu + \mu^2)}}{N}\\ &= \frac{\sum\limits_{i=1}^N{x_i^2} - 2\mu\sum\limits_{i=1}^N{x_i} + \mu^{2}N}{N}\\ &= \frac{\sum\limits_{i=1}^N{x_i^2}}{N} - 2\mu^2 + \mu^2\\ &= \frac{\sum\limits_{i=1}^N{x_i^2}}{N} - \mu^2\\ &= \frac{\sum\limits_{i=1}^N{x_i}}{N} - \frac{(\sum\limits_{i=1}^N{x_i})^2}{N^2} \end{align}\]
1.4 Sample Variance
\[\bar{x} = \frac{\sum\limits_{i=1}^n{x_i}}{n}\]
\[{S_n}^2 = \frac{\sum\limits_{i=1}^n{(x_i - \bar{x})^2}}{n}\]
1.5 Compare Population With Sample
| Concept | Population | Sample |
|---|---|---|
| Mean | \[\mu = \frac{\sum\limits_{i=1}^N{x_i}}{N} = \frac{x_1 + x_2 + … + x_N}{N}\] | \[\bar{x} = \frac{\sum\limits_{i=1}^n{x_i}}{n}\] |
| Variance | \[\sigma ^2 = \frac{\sum\limits_{i=1}^N{(x_i - \mu)}}{N}\] | \[{S_n}^2 = \frac{\sum\limits_{i=1}^n{(x_i - \bar{x})^2}}{n-1}\] |
| Standard Deviation | \[\sigma = \sqrt{\sigma^2} = \sqrt{\frac{\sum\limits_{i=1}^N{(x_i - \mu)}}{N}}\] | \[S = \sqrt{S^2}\] |
1.6 Random Variable
A function that maps you from the world of random processes to an actual number.
2. 基础分布
2.1 Uniform Distribution
All of the outcomes are equally likely.
2.2 Binomial Distribution
\[\begin{align} E(X) &= \sum\limits_{k=1}^n{k\binom{n}{k}p^k(1-p)^{n-k}}\\ &= \sum\limits_{k=1}^n{k\frac{n!}{k!(n-k)!}p^k(1-p)^{n-k}}\\ &= \sum\limits_{k=1}^n{\frac{n!}{(k-1)!(n-k)!}p^k(1-p)^{n-k}}\\ &= np\sum\limits_{k=1}^n{\frac{(n-1)!}{(k-1)!(n-k)!}p^{k-1}(1-p)^{n-k}}\\ &= np\sum\limits_{k=0}^{n-1}{\binom{n-1}{k}p^k(1-p)^{n-1-k}}\\ &= np \end{align}\]
2.3 Poisson Distribution
\[E(X) = \lambda = np\]
\[\begin{align} P(x=k) &= \lim\limits_{n \to \infty}{\binom{n}{k}{(\frac{\lambda}{k})^2}(1-\frac{\lambda}{n})^{n-k}}\\ &= \frac{\lambda^k}{k!}e^{-\lambda} \end{align}\]
2.4 Gaussian Distribution
\[\begin{align} P(x) &= \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2} \end{align}\]