一. 随机变量
1.1 随机变量概述
随机变量表示结果是随机的,例如骰子的点数,是1-6之间的一个随机数。
![](https://img.haomeiwen.com/i2638478/564e6d2c74370d30.png)
例子:
![](https://img.haomeiwen.com/i2638478/cbd6e1f482c21aff.png)
1.2 离散 or 连续
![](https://img.haomeiwen.com/i2638478/53c6cac921d09e72.png)
1.3 取值概率
![](https://img.haomeiwen.com/i2638478/650d23ba84e58548.png)
二. 随机变量分布
2.1 分布律
![](https://img.haomeiwen.com/i2638478/d275ca0c11e1b71b.png)
例1:
![](https://img.haomeiwen.com/i2638478/d07c356fec6e85d8.png)
例2:
![](https://img.haomeiwen.com/i2638478/0069195bcdaa4253.png)
2.2 (0-1)分布
从例子中看(0-1)分布:
![](https://img.haomeiwen.com/i2638478/eae27171d31e4051.png)
常见(0-1) 分布:
![](https://img.haomeiwen.com/i2638478/f0b9ab62b6dc96a9.png)
2.3 伯努利试验
![](https://img.haomeiwen.com/i2638478/cc92978a8a5577c4.png)
n重伯努利试验
![](https://img.haomeiwen.com/i2638478/f5635386cee6ce5c.png)
2.4 二项分布
![](https://img.haomeiwen.com/i2638478/3378d535266fc020.png)
![](https://img.haomeiwen.com/i2638478/9da3327834a2d785.png)
二项分布的定义:
![](https://img.haomeiwen.com/i2638478/a17a0c7ab9816af3.png)
例1:
![](https://img.haomeiwen.com/i2638478/b849db2b74f91ed7.png)
例2:
![](https://img.haomeiwen.com/i2638478/ebb91e68e2ea6070.png)
![](https://img.haomeiwen.com/i2638478/80677f3b7eef8130.png)
2.5 泊松分布
![](https://img.haomeiwen.com/i2638478/32541197b6ae8451.png)
泊松分布的近似二项分布:
![](https://img.haomeiwen.com/i2638478/2fa7a2c56a26d383.png)
例1:
![](https://img.haomeiwen.com/i2638478/7860978bf23813fc.png)
2.6 分布函数
![](https://img.haomeiwen.com/i2638478/6c9312bab660db62.png)
案例:
![](https://img.haomeiwen.com/i2638478/19afd6490e47e26c.png)
书本上例1:
![](https://img.haomeiwen.com/i2638478/2c1d62312cbe9328.png)
![](https://img.haomeiwen.com/i2638478/1094cfef2f3e8502.png)
2.7 连续性随机变量的分布函数
![](https://img.haomeiwen.com/i2638478/40d1cda5cffbbd61.png)
![](https://img.haomeiwen.com/i2638478/01f792ac0c64468c.png)
概率密度函数:
![](https://img.haomeiwen.com/i2638478/3c994c780e437f4b.png)
连续性随机变量,取值非常多,所以不是取某个点的概率,而是一个累积的概率,例如 x<=100的概率
概率密度:
![](https://img.haomeiwen.com/i2638478/49ac3ed0da412046.png)
例子:
![](https://img.haomeiwen.com/i2638478/f9f82a1f9da6ae22.png)
2.8 二项分布和正态分布
![](https://img.haomeiwen.com/i2638478/ef83f8151a816d0e.png)
正态分布:
![](https://img.haomeiwen.com/i2638478/2db41d768cc92247.png)
![](https://img.haomeiwen.com/i2638478/dba5436731071f53.png)
![](https://img.haomeiwen.com/i2638478/7fd37ca2081e0842.png)
![](https://img.haomeiwen.com/i2638478/bef09739832557f5.png)
正态分布->标准正态分布
![](https://img.haomeiwen.com/i2638478/7d8d32e0ddbaa918.png)
例子:
![](https://img.haomeiwen.com/i2638478/ba86172e7f0dfa0b.png)
2.9 均匀分布
![](https://img.haomeiwen.com/i2638478/98d88d1be0d019be.png)
![](https://img.haomeiwen.com/i2638478/7b4d5de6d0ac930a.png)
例子:
![](https://img.haomeiwen.com/i2638478/275840442241ec14.png)
2.10 微积分简单的教程
学统计学,离不开微积分,下面是微积分简单的教程。
文字版:
https://zhuanlan.zhihu.com/p/481027124
视频版:
https://www.bilibili.com/video/BV1if4y187mK?spm_id_from=333.999.0.0
https://www.bilibili.com/video/BV1KM4y1g7ud?spm_id_from=333.999.0.0
https://www.bilibili.com/video/BV1PL411W7Mu?spm_id_from=333.999.0.0
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