常见统计分布的概率分布图
⽂章⽬录
import  numpy as  np
import  matplotlib .pyplot as  plt
import  seaborn as  sns
from  scipy import  stats
sns .set ()
sns .set_style ('ticks')
sns .set_context ('talk')
%matplotlib inline
Normal distribution
x = np .linspace (-5, 5, 40)
plt .figure (figsize =(14, 5))
# probability density function
plt .subplot (121)
for  sigma in  (0.4, 1, 1.4):
plt .plot (x , stats .norm .pdf (x , loc =0, scale =sigma ), label =f '$\sigma={sigma}$')
plt .legend ()
sns .despine ()
plt .title ('pdf of normal distribution') #fontdict={'fontsize':16}
# cumulative distribution function
plt .subplot (122)
for  sigma in  (0.4, 1, 1.4):
plt .plot (x , stats .norm .cdf (x , loc =0, scale =sigma ), label =f '$\sigma={sigma}$')
plt .legend ()
sns .despine ()
plt .title ('cdf of normal distribution') #fontdict={'fontsize':16})
plt .show
()
distribution
plt .figure (figsize =(15, 5.5))
plt .subplot (121)
x = np .linspace (0.2, 12, 2000)
for  k in  range (1, 6, 1):
plt .plot (x , stats .chi2.pdf (x , df =k ), label =f 'df={k}')
plt .legend ()
sns .despine ()
plt .subplot (122)
x = np .linspace (0, 50, 2000)
for  k in  range (5, 30, 5):
plt .plot (x , stats .chi2.pdf (x , df =k ), label =f 'df={k}')
plt .legend ()
sns .despine ()
plt .suptitle ('pdf of chi2-square distribution', fontsize =18)
plt .show
()
χ2f (x ,k )=x exp −x /22Γk /2k /2()1
k /2−1()
plt .figure (figsize =(15, 5.5))
plt .subplot (121)
x = np .linspace (0.2, 12, 2000)
for  k in  range (1, 6, 1):
plt .plot (x , stats .chi2.cdf (x , df =k ), label =f 'df={k}')
plt .legend ()
sns .despine ()
plt .subplot (122)
x = np .linspace (0, 50, 2000)
for  k in  range (5, 30, 5):
plt .plot (x , stats .chi2.cdf (x , df =k ), label =f 'df={k}')
plt .legend ()
sns .despine ()
plt .suptitle ('cdf of chi2-square distribution', fontsize =18)
plt .show
()
t-distribution
plt .figure (figsize =(7, 5))
x = np .linspace (-5, 5)
for  k in  range (1, 8, 2):
plt .plot (x , stats .t .pdf (x , df =k ), label =f 'df={k}')
plt .legend ()
sns .despine ()
plt .title ('pdf of t distribution')
plt .show
()
plt .figure (figsize =(7, 5))
x = np .linspace (-5, 5)
for  k in  range (1, 8, 2):
plt .plot (x , stats .t .cdf (x , df =k ), label =f 'df={k}')
plt .legend ()
sns .despine ()
plt .title ('cdf of t distribution')
plt .show
()
F-distribution
f (x ,ν)=(1+Γ(ν)πνΓ((ν+1)/2)
x /ν)2−(ν+1)/2
x = np.linspace(0.05,5,400)
plt.figure(figsize=(15,5.5))
plt.subplot(121)
for d1, d2 in[(1,1),(3,1),(5,2),(10,2),(10,10)]:
y = stats.f.pdf(x, d1, d2)
plt.ylim(-0.05,1.75)
plt.plot(x, y, label=f'd1={d1}, d2={d2}')
plt.legend()
sns.despine()
plt.subplot(122)
for d1, d2 in[(1,1),(1,2),(2,5),(1,10),(30,10)]:
y = stats.f.pdf(x, d1, d2)
plt.plot(x, y, label=f'd1={d1}, d2={d2}')
plt.legend()
sns.despine()
plt.suptitle('pdf of F distribution', fontsize=18)
plt.show()
x = np.linspace(0.05,5,400)
plt.figure(figsize=(15,5.5))
plt.subplot(121)
linspace numpyfor d1, d2 in[(1,1),(3,1),(5,2),(10,2),(10,10)]:
y = stats.f.cdf(x, d1, d2)
plt.plot(x, y, label=f'd1={d1}, d2={d2}')
plt.legend()
sns.despine()
plt.subplot(122)
for d1, d2 in[(1,1),(1,2),(2,5),(1,10),(30,10)]:
y = stats.f.cdf(x, d1, d2)
plt.plot(x, y, label=f'd1={d1}, d2={d2}')
plt.legend()
sns.despine()
plt.suptitle('cdf of F distribution', fontsize=18)
plt.show()
Beta distribution
x = np.linspace(0.05,0.95,400)
plt.figure(figsize=(15,6))
plt.subplot(121)
for alpha, beta in[(0.5,0.5),(5,1),(1,3),(1,1),(2,5)]:
plt.plot(x, stats.beta.pdf(x, alpha, beta), label=r'$\alpha={}, \beta={}$'.format(alpha, beta)) plt.legend()
sns.despine()
plt.title('pdf of Beta distribution')
plt.subplot(122)
for alpha, beta in[(0.5,0.5),(5,1),(1,3),(2,2),(2,5)]:
plt.plot(x, stats.beta.cdf(x, alpha, beta), label=r'$\alpha={}, \beta={}$'.format(alpha, beta)) plt.legend()
sns.despine()
plt.title('cdf of Beta distribution')
plt.show()
对中⼼极限定理的验证
。
⽬标是考察  的分布情况
以参数为 1 的指数分布为例
sns .set_context ('notebook')
plt .figure (figsize =(15, 15))
for  i in  range (1, 17):
plt .subplot (4, 4, i )
n = 4 * i
rvs = [
(np .random .exponential (size =n ).sum () - n ) / np .sqrt (n )
for  _ in  range (10000)
]
sns .distplot (rvs , kde =False )
plt .title (f "$n={n}$");sns .despine ()
plt .show ()
参数为 1 的泊松分布，这是⼀个离散型的分布
plt .figure (figsize =(15, 15))
for  i in  range (1, 17):
plt .subplot (4, 4, i )
n = 20 * i
rvs = [
(np .random .poisson (size =n ).sum () - n ) / np .sqrt (n )
for  _ in  range (5000)
]
sns .distplot (rvs , kde =False )
plt .title (f "$n={n}$");sns .despine ()
plt .show ()
从上⾯可以看到，对于⼀个连续型的分布，它的“标准型”能够更快的趋近于正态分布。⽽对于离散型分布⽽⾔，它所需要的  可能就⽐较⼤了。ϵ=n →nσ2S −nμ
n d N (0,1)ϵ,ϵ,ϵ,⋯,ϵ,ϵ..234100.n

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