python 计算概率密度、累计分布、逆函数的例子

>>> import scipy.stats as st

>>> st.norm.cdf(0) # 标准正态分布在 0 处的累计分布概率值
0.5

>>> st.norm.cdf([-1, 0, 1])# 标准正态分布分别在 -1， 0， 1 处的累计分布概率值
array([0.15865525, 0.5, 0.84134475])

>>> st.norm.pdf(0) # 标准正态分布在 0 处的概率密度值
0.3989422804014327

>>> st.norm.ppf(0.975)# 标准正态分布在 0.975 处的逆函数值
1.959963984540054

>>> st.norm.lsf(0.975)# 标准正态分布在 0.025 处的生存函数的逆函数值
1.959963984540054

>>> st.norm.cdf(0, loc=2, scale=1) # 均值为 2，标准差为 1 的正态分布在 0 处的累计分布概率值
0.022750131948179195

>>> st.binom.pmf(4, n=100, p=0.05) # 参数值 n=100, p=0.05 的二项分布在 4 处的概率密度值
0.17814264156968956

>>> st.geom.pmf(4, p=0.05) # 参数值 p=0.05 的几何分布在 4 处的概率密度值
0.04286875

>>> st.poisson.pmf(2, mu=3) # 参数值 mu=3 的泊松分布在 2 处的概率密度值
0.22404180765538775

>>> st.chi2.ppf(0.95, df=10) # 自由度为 10 的卡方分布在 0.95 处的逆函数值
18.307038053275146

>>> st.t.ppf(0.975, df=10) # 自由度为 10 的 t 分布在 0.975 处的逆函数值
2.2281388519649385

>>> st.f.ppf(0.95, dfn=2, dfd=12) # 自由度为 2, 12 的 F 分布在 0.95 处的逆函数值
3.8852938346523933

import numpy as np
import math
import random
import matplotlib.pyplot as plt
import collections

count_dict = dict()
bin_count = 20

def inverseCDF():
 """
 return the x value in PDF
 """
 uniform_random = random.random()
 return inverse_cdf(uniform_random)
 

def pdf(x):
 return 2 * x
 
# cdf = x^2, 其逆函数很好求，因此直接用公式法
def inverse_cdf(x):
 return math.sqrt(x)


def draw_pdf(D):
	global bin_count
 D = collections.OrderedDict(sorted(D.items()))
 plt.bar(range(len(D)), list(D.values()), align='center')
 # 因为映射bin的时候采用的floor操作，因此加上0.5
 value_list = [(key + 0.5) / bin_count for key in D.keys()]
 plt.xticks(range(len(D)), value_list)
 plt.xlabel('x', fontsize=5)
 plt.ylabel('counts', fontsize=5)
 plt.title('counting bits')
 plt.show()

for i in range(90000):
 x = inverseCDF()
 # 用bin去映射，否则不好操作
 bin = math.floor(x * bin_count) # type(bin): int
 count_dict[bin] = count_dict.get(bin, 0) + 1

draw_pdf(count_dict)

import numpy as np
import math
import random
import matplotlib.pyplot as plt
import collections

lookup_table_size = 40
CDFlookup_table = np.zeros((lookup_table_size))

count_dict = dict()
bin_count = 20

def inverse_cdf_numerically(y):
 global lookup_table_size
 global CDFlookup_table
 value = 0.0
 for i in range(lookup_table_size):
  x = i * 1.0 / (lookup_table_size - 1)
  value += pdf2(x)
  CDFlookup_table[i] = value
 CDFlookup_table /= value # normalize the cdf

 if y < CDFlookup_table[0]: 
  t = y / CDFlookup_table[0]
  return t / lookup_table_size
 index = -1
 for j in range(lookup_table_size):
  if CDFlookup_table[j] >= y:
   index = j
   break
 # linear interpolation
 t = (y - CDFlookup_table[index - 1]) / \
  (CDFlookup_table[index] - CDFlookup_table[index - 1])
 fractional_index = index + t # 因为index从0开始,所以不是 (index-1)+t
 return fractional_index / lookup_table_size


def inverseCDF():
 """
 return the x value in PDF
 """
 uniform_random = random.random()
 return inverse_cdf_numerically(uniform_random)


def pdf2(x):
 return (x * x * x - 10.0 * x * x + 5.0 * x + 11.0) / (10.417)

def draw_pdf(D):
 global bin_count
 D = collections.OrderedDict(sorted(D.items()))
 plt.bar(range(len(D)), list(D.values()), align='center')
 value_list = [(key + 0.5) / bin_count for key in D.keys()]
 plt.xticks(range(len(D)), value_list)
 plt.xlabel('x', fontsize=5)
 plt.ylabel('counts', fontsize=5)
 plt.title('counting bits')
 plt.show()


for i in range(90000):
 x = inverseCDF()
 bin = math.floor(x * bin_count) # type(bin): int
 count_dict[bin] = count_dict.get(bin, 0) + 1

draw_pdf(count_dict)

import numpy as np
import matplotlib.pyplot as plt
"""
reference:
https://blog.demofox.org/2017/07/25/counting-bits-the-normal-distribution/
"""


def plot_bar_x():
 # this is for plotting purpose
 index = np.arange(counting.shape[0])
 plt.bar(index, counting)
 plt.xlabel('x', fontsize=5)
 plt.ylabel('counts', fontsize=5)
 plt.title('counting bits')
 plt.show()


# if dice_side=2, is binomial distribution
# if dice_side>2 , is multinomial distribution
dice_side = 2
# if N becomes larger, then multinomial distribution will more like normal distribution
N = 100

counting = np.zeros(((dice_side - 1) * N + 1))

for i in range(30000):
 sum = 0
 for j in range(N):
  dice_result = np.random.randint(0, dice_side)
  sum += dice_result

 counting[sum] += 1

# normalization
counting /= np.sum(counting)
plot_bar_x()