如何计算python中正态累积分布函数的反函数？

Question 1

如何计算Python中正态分布的累积分布函数（CDF）的反函数？

我应该使用哪个库？可能是卑鄙的？

Question 2

NORMSINV（在注释中提到）是标准正态分布的CDF的倒数。使用scipy，您可以使用对象的ppf方法进行计算scipy.stats.norm。首字母缩写词ppf代表百分点函数，它是分位数函数的另一个名称。

In [20]: from scipy.stats import norm

In [21]: norm.ppf(0.95)
Out[21]: 1.6448536269514722

检查它是否与CDF相反：

In [34]: norm.cdf(norm.ppf(0.95))
Out[34]: 0.94999999999999996

默认情况下，norm.ppf使用mean = 0和stddev = 1，这是“标准”正态分布。您可以通过分别指定loc和scale参数来使用不同的均值和标准差。

In [35]: norm.ppf(0.95, loc=10, scale=2)
Out[35]: 13.289707253902945

如果查看源代码scipy.stats.norm，您会发现该ppf方法最终会调用scipy.special.ndtri。因此，要计算标准正态分布的CDF的倒数，可以直接使用该函数：

In [43]: from scipy.special import ndtri

In [44]: ndtri(0.95)
Out[44]: 1.6448536269514722

Question 3

# given random variable X (house price) with population muy = 60, sigma = 40
import scipy as sc
import scipy.stats as sct
sc.version.full_version # 0.15.1

#a. Find P(X<50)
sct.norm.cdf(x=50,loc=60,scale=40) # 0.4012936743170763

#b. Find P(X>=50)
sct.norm.sf(x=50,loc=60,scale=40) # 0.5987063256829237

#c. Find P(60<=X<=80)
sct.norm.cdf(x=80,loc=60,scale=40) - sct.norm.cdf(x=60,loc=60,scale=40)

#d. how much top most 5% expensive house cost at least? or find x where P(X>=x) = 0.05
sct.norm.isf(q=0.05,loc=60,scale=40)

#e. how much top most 5% cheapest house cost at least? or find x where P(X<=x) = 0.05
sct.norm.ppf(q=0.05,loc=60,scale=40)

Question 4

Starting Python 3.8, the standard library provides the NormalDist object as part of the statistics module.

It can be used to get the inverse cumulative distribution function (inv_cdf - inverse of the cdf), also known as the quantile function or the percent-point function for a given mean (mu) and standard deviation (sigma):

from statistics import NormalDist

NormalDist(mu=10, sigma=2).inv_cdf(0.95)
# 13.289707253902943

Which can be simplified for the standard normal distribution (mu = 0 and sigma = 1):

NormalDist().inv_cdf(0.95)
# 1.6448536269514715