有名確率分布18種類のチートシートとPythonによる可視化

有名確率分布18種類のチートシート

分布名	平均	分散	形状	一言メモ
Normal	$$\mu$$	$$\sigma^2$$	Continuous	よく使われる"ベル曲線"
Binomial	$$n \times p$$	$$n \times p \times (1 - p)$$	Discrete	成功回数の分布
Poisson	$$\lambda$$	$$\lambda$$	Discrete	希なイベントの回数
Exponential	$$1/\lambda$$	$$1/\lambda^2$$	Continuous	イベント間の待ち時間
Uniform	$$(a + b) / 2$$	$$(b - a)^2 / 12$$	Continuous	等確率で値が発生
Gamma	$$\alpha \times \beta$$	$$\alpha \times \beta^2$$	Continuous	イベントの発生までの時間
Beta	$$\alpha / (\alpha + \beta)$$	$$\alpha \beta / ((\alpha + \beta)^2 (\alpha + \beta + 1))$$	Continuous	確率の確率
Geometric	$$1/p$$	$$(1-p)/p^2$$	Discrete	初めて成功するまでの試行数
Negative Binomial	$$r/p$$	$$r(1-p)/p^2$$	Discrete	$$r$$回成功するまでの試行数
Hypergeometric	$$N_1n/N$$	$$N_1N_2n(N-n)/(N^2(N-1))$$	Discrete	無戻し抽出
Bernoulli	$$p$$	$$p(1-p)$$	Discrete	成功か失敗
Log-Normal	$$e^{(\mu+\sigma^2/2)}$$	$$(e^{\sigma^2}-1)e^{2\mu+\sigma^2}$$	Continuous	正の値、対数が正規分布
Weibull	$$\beta \Gamma(1+1/\alpha)$$	$$\beta^2[\Gamma(1+2/\alpha) - (\Gamma(1+1/\alpha))^2]$$	Continuous	耐久性・寿命
Cauchy	Undefined	Undefined	Continuous	平均・分散なし
Pareto	$$\alpha x_m / (\alpha - 1)$$ for $$\alpha > 1$$	$$\alpha x_m^2 / ((\alpha - 1)^2 (\alpha - 2))$$ for $$\alpha > 2$$	Continuous	"80-20"ルール
Chi-Squared	$$k$$	$$2k$$	Continuous	自由度$$k$$
Student's t	$$0$$ if $$\nu > 1$$	$$\nu/(\nu - 2)$$ if $$\nu > 2$$	Continuous	正規分布の代わりに使われる
F-distribution	$$\nu_2 (\nu_1 - 2) / (\nu_1 (\nu_2 - 2))$$ for $$\nu_1 > 2, \nu_2 > 4$$	Varies	Continuous	分散比のテスト

Pythonによる可視化

# Import necessary libraries
import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats

# Create subplots
fig, axs = plt.subplots(6, 3, figsize=(18, 24))

# List of distributions to plot
distributions = [
    ('Normal', stats.norm(loc=0, scale=1)),
    ('Binomial', stats.binom(n=10, p=0.5)),
    ('Poisson', stats.poisson(mu=3)),
    ('Exponential', stats.expon(scale=1)),
    ('Uniform', stats.uniform(loc=0, scale=10)),
    ('Gamma', stats.gamma(a=2, scale=1)),
    ('Beta', stats.beta(a=2, b=5, loc=0, scale=1)),
    ('Geometric', stats.geom(p=0.5)),
    ('Negative Binomial', stats.nbinom(n=10, p=0.5)),
    ('Hypergeometric', stats.hypergeom(M=20, n=7, N=12)),
    ('Bernoulli', stats.bernoulli(p=0.6)),
    ('Log-Normal', stats.lognorm(s=0.954, loc=0, scale=np.exp(0.65))),
    ('Weibull', stats.weibull_min(c=1.79, loc=0, scale=1)),
    ('Cauchy', stats.cauchy(loc=0, scale=1)),
    ('Pareto', stats.pareto(b=2.62)),
    ('Chi-Squared', stats.chi2(df=5)),
    ('Student\'s t', stats.t(df=10)),
    ('F-distribution', stats.f(dfn=29, dfd=18))
]

# Plot each distribution
for ax, (name, dist) in zip(axs.flatten(), distributions):
    if name == 'Bernoulli':
        x = np.array([0, 1])
        ax.bar(x, dist.pmf(x), alpha=0.7, label='pmf')
    elif name in ['Binomial', 'Poisson', 'Geometric', 'Negative Binomial', 'Hypergeometric']:
        x = np.arange(0, 11)
        ax.bar(x, dist.pmf(x), alpha=0.7, label='pmf')
    else:
        x = np.linspace(dist.ppf(0.01), dist.ppf(0.99), 100)
        ax.plot(x, dist.pdf(x), label='pdf')
    ax.set_title(name)

# Adjust layout to prevent overlap
plt.tight_layout()

# Show the plot
plt.show()