【常见分布及其特征(8)】连续型随机变量-正态分布*

正态分布

引例一

1. 定义随机变量
   定义了一个连续均匀分布$U(0,1)$,并且定义了$n=5$个独立的随机变量$X_1$到$X_5$;并且均服从上述分布;即$$X_n\sim U(0,1),n=1,2,3,4,5$$
   则可以称这5个随机变量是独立同分布的

2. 抽样
   对5个随机变量执行一次抽样;
   样本值：用小写字母表示，例如一次采样后得到的具体值为 $x_1,x_2,\dots ,x_5$,其中每个 $x_i\in [0,1]$。一次采样后，样本值为:
   $$x_1=\text{Sample}(X_1),x_2=\text{Sample}(X_2),\cdots ,x_5=\text{Sample}(X_5)$$
   以下是可能的抽样结果示例,这样的示例可以举出无限多个:

示例编号	均值	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$
示例 1	0.549289	0.896074	0.886996	0.374390	0.081029	0.507958
示例 2	0.419763	0.570370	0.107197	0.983434	0.221526	0.216287
示例 3	0.294898	0.185088	0.013277	0.408898	0.367660	0.499566

附一个生成的代码

import numpy as np

sp = np.random.random((3,5))
print(sp)
# 对每行求均值
row_means = np.mean(sp, axis=1)
print(row_means)
print(np.column_stack((row_means, sp)))

3. 扩大随机变量,再次抽样
   $n=1000$个独立的随机变量$X_1$到$X_{1000}$;并且均服从上述分布;即$$X_n\sim U(0,1),n=1,2,3,\cdots ,1000$$

示例编号	均值	$x_1$	$x_2$	$x_3$	$\cdots$	$x_{1000}$
示例 1	0.50698297	0.84908216	0.13467754	0.374390	$\cdots$	0.06873601
示例 2	0.48454968	0.66022503	0.03217824	0.983434	$\cdots$	0.70057909
示例 3	0.49077309	0.25737745	0.04264012	0.408898	$\cdots$	0.30889889

观察均值这一列,可以明显看出,当$n=5\to n=1000$,时,均值的稳定程度明显提高了,而且趋近于$X_i$的期望了;

• 如果觉得以上的不够直观,以下展示了各12组示例,第一列均为均值,均匀分布;

# n=5时的结果
[0.51177121 0.64972725 0.79864437 0.26144494 0.02885988 0.82017961]
[0.5387426  0.15816605 0.72253606 0.67549004 0.60952049 0.52800036]
[0.25243907 0.87643116 0.09397123 0.00179747 0.02647162 0.26352389]
[0.49120919 0.37110317 0.56440969 0.69843874 0.38633834 0.43575599]
[0.49709079 0.96020927 0.6040899  0.60171813 0.05464104 0.26479562]
[0.56874528 0.15495668 0.72611065 0.6985964  0.61305459 0.65100806]
[0.56970291 0.21338898 0.72839761 0.26438999 0.98735146 0.6549865 ]
[0.57184459 0.56795882 0.53453043 0.59070316 0.75968399 0.40634656]
[0.67773425 0.91024391 0.85876042 0.70395502 0.10358019 0.81213171]
[0.64856283 0.57525252 0.87790623 0.49302471 0.31732044 0.97931024]
[0.37102274 0.65237646 0.55096309 0.44173061 0.07955758 0.13048595]
[0.4822816  0.47407257 0.21329814 0.27420517 0.98064111 0.46919102]

# n=1000时的结果
[0.4945074  0.50273574 0.52522912 ... 0.38031662 0.42860546 0.72074343]
[0.49621251 0.67881337 0.03880736 ... 0.92981839 0.65805523 0.38261631]
[0.49262885 0.81203807 0.43174388 ... 0.68740689 0.92866097 0.56505152]
[0.50967169 0.14997624 0.93324006 ... 0.53000919 0.48711552 0.26967198]
[0.49939796 0.63793338 0.87528609 ... 0.98436646 0.91608776 0.32658811]
[0.50291407 0.32440555 0.03177322 ... 0.96926983 0.52440474 0.73910828]
[0.50908082 0.88887095 0.46454131 ... 0.04160692 0.63408981 0.72006609]
[0.5006352  0.25307945 0.5568652  ... 0.96087729 0.10194726 0.54255956]
[0.49424821 0.08349161 0.35183987 ... 0.22038619 0.93989811 0.14642372]
[0.48437207 0.4528952  0.91680597 ... 0.20512634 0.19849493 0.53023536]
[0.49658921 0.92238767 0.17093952 ... 0.71290388 0.41343221 0.18931387]
[0.49429535 0.96498261 0.81471275 ... 0.93917227 0.84554065 0.59315527]

• 不同的$n$绘制分布图像如下,可见随着n的增大,图像趋向于一个钟形曲线;
  

4. 其他分布抽样
   若随机变量服从指数分布呢?设$X_i\sim \text{Exp}(2)$
   执行相同的采样方式,可见,同样伴随着n的增大,$均值$这一列的稳定性提高了,且都趋向于$X_i$的期望

# n=5时的结果
[0.73411466 0.70151055 0.58552744 1.40848118 0.52992534 0.44512879]
[0.81113348 0.53431281 0.27084777 0.64021452 0.17419464 2.43609766]
[0.40670354 0.28069173 0.09011335 0.46911745 0.77372054 0.4198746 ]
[0.39217388 0.71925745 0.33419681 0.14974111 0.43873465 0.31893941]
[1.10752828 0.91971051 1.88885734 0.78719437 1.72043777 0.22144144]
[0.66299886 0.53282981 0.52905482 0.51776715 1.00682295 0.72851959]
[0.52414378 0.55970346 1.30554564 0.35656776 0.16566018 0.23324187]
[0.74790822 0.15209213 1.45643592 0.00856253 1.65698463 0.4654659 ]
[0.47633977 0.08690879 0.7636792  0.09175103 0.91023342 0.52912641]
[0.24892092 0.11581998 0.58664384 0.07320079 0.35082388 0.11811614]
[0.76317128 0.18174787 1.81766939 0.37271795 0.9385246  0.50519659]
[0.28458274 0.54393585 0.21895033 0.05139447