機率分布 (Probability Distributions)

套路 4: 機率分布 (Probability Distributions)

什麼是資料的機率分布? 說白了就是描述不同結果可能發生的機率的數學函數 (probability density function，pdf)。以下是舉例使用Python模擬幾種機率分布並畫圖。

Normal distribution公式 (probability density function):

其中μ是母體(population)的算數平均數，s是母體標準差 (population standard deviation)，p與e是常數數。由此公式可知常態分佈由兩個參數(μ, s )決定。

(μ = 0, s = 1 ) 平均值為0、標準差為1的標準常態分布(standard normal distribution) 。

scipy.stats.norm程式範例: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.norm

from scipy.stats import norm

import numpy as np

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 1)

mean, var, skew, kurt = norm.stats(moments='mvsk')

x = np.linspace(norm.ppf(0.01), norm.ppf(0.99), 100)   # ppf: percent point function (percentiles)

ax.plot(x, norm.pdf(x), 'r-', lw=5, alpha=0.6, label='norm pdf')   # pdf: probability density function

rv = norm()

ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

r = norm.rvs(size=1000)

ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)

ax.legend(loc='best', frameon=False)

plt.show()

結果: 平均值為0、標準差為1的標準常態分布(standard normal distribution) 。

t distribution公式 (probability density function):

其中n是自由度 (degree of freedom) ，G是gamma function。由此公式可知t分佈由參數n決定。

scipy.stats.t程式範例: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.t.html#scipy.stats.t

from scipy.stats import t

import numpy as np

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 1)

df = 2.74

mean, var, skew, kurt = t.stats(df, moments='mvsk')

x = np.linspace(t.ppf(0.01, df), t.ppf(0.99, df), 100)

ax.plot(x, t.pdf(x, df), 'r-', lw=5, alpha=0.6, label='t pdf')

rv = t(df)

ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

vals = t.ppf([0.001, 0.5, 0.999], df)

np.allclose([0.001, 0.5, 0.999], t.cdf(vals, df))

r = t.rvs(df, size=1000)

ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)

ax.legend(loc='best', frameon=False)

plt.show()

結果: df = 2.74的t分布

F distribution公式 (probability density function):

其中d₁ d₂是自由度 (degree of freedom) ，B是beta function。由此公式可知F分佈由參數d₁ d₂決定。

scipy.stats.f程式範例: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.f.html#scipy.stats.f

from scipy.stats import f

import numpy as np

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 1)

dfn, dfd = 29, 18

mean, var, skew, kurt = f.stats(dfn, dfd, moments='mvsk')

x = np.linspace(f.ppf(0.01, dfn, dfd), f.ppf(0.99, dfn, dfd), 100)

ax.plot(x, f.pdf(x, dfn, dfd), 'r-', lw=5, alpha=0.6, label='f pdf')

rv = f(dfn, dfd)

ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

vals = f.ppf([0.001, 0.5, 0.999], dfn, dfd)

np.allclose([0.001, 0.5, 0.999], f.cdf(vals, dfn, dfd))

r = f.rvs(dfn, dfd, size=1000)

ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)

ax.legend(loc='best', frameon=False)

plt.show()

結果: df = 29, 18的F分布

c² (Chi-square) distribution公式 (probability density function):

其中k是自由度 (degree of freedom) ，G是gamma function。由此公式可知c²分佈由參數k決定。

scipy.stats.chi2程式範例: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.chi2.html#scipy.stats.chi2

from scipy.stats import chi2

import numpy as np

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 1)

df = 55

mean, var, skew, kurt = chi2.stats(df, moments='mvsk')

x = np.linspace(chi2.ppf(0.01, df), chi2.ppf(0.99, df), 100)

ax.plot(x, chi2.pdf(x, df), 'r-', lw=5, alpha=0.6, label='chi2 pdf')

rv = chi2(df)

ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

vals = chi2.ppf([0.001, 0.5, 0.999], df)

np.allclose([0.001, 0.5, 0.999], chi2.cdf(vals, df))

r = chi2.rvs(df, size=1000)

ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)

ax.legend(loc='best', frameon=False)

plt.show()

結果: df = 55的c²分布

Uniform distribution:

scipy.stats.uniform程式範例: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.uniform.html#scipy.stats.uniform

from scipy.stats import uniform

import numpy as np

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 1)

mean, var, skew, kurt = uniform.stats(moments='mvsk')

x = np.linspace(uniform.ppf(0.01), uniform.ppf(0.99), 100)

ax.plot(x, uniform.pdf(x), 'r-', lw=5, alpha=0.6, label='uniform pdf')

rv = uniform()

ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

vals = uniform.ppf([0.001, 0.5, 0.999])

np.allclose([0.001, 0.5, 0.999], uniform.cdf(vals))

r = uniform.rvs(size=1000)

ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)

ax.legend(loc='best', frameon=False)

plt.show()

結果: uniform分布

 

Binomial distribution公式 (probability density function):

 

其中p (1 - p)是事件出現的機率， k Î {0, 1,..., n}。由此公式可知二項式分佈由參數p及n決定。

scipy.stats.binom程式範例: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.binom.html#scipy.stats.binom

from scipy.stats import binom

import numpy as np

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 1)

n, p = 5, 0.4

mean, var, skew, kurt = binom.stats(n, p, moments='mvsk')

x = np.arange(binom.ppf(0.01, n, p), binom.ppf(0.99, n, p))

ax.plot(x, binom.pmf(x, n, p), 'bo', ms=8, label='binom pmf')

ax.vlines(x, 0, binom.pmf(x, n, p), colors='b', lw=5, alpha=0.5)

rv = binom(n, p)

ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1, label='frozen pmf')

ax.legend(loc='best', frameon=False)

plt.show()

結果: n, p = 5, 0.4的二項式分布

Poisson distribution公式 (probability density function):

其中l是事件出現機率的期望值， k Î {0, 1,..., n}。由此公式可知布ㄚ松分佈由參數l決定。

scipy.stats.poisson程式範例: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson

from scipy.stats import poisson

import numpy as np

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 1)

mu = 0.6

mean, var, skew, kurt = poisson.stats(mu, moments='mvsk')

x = np.arange(poisson.ppf(0.01, mu), poisson.ppf(0.99, mu))

ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf')

ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5)

rv = poisson(mu)

ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1, label='frozen pmf')

ax.legend(loc='best', frameon=False)

plt.show()

結果: l = 0.6的布ㄚ松分佈