二项式分布（Binomial Distribution）-CFANZ编程社区

定义

让我们来看看玩板球这个例子。假设你今天赢了一场比赛，这表示一个成功的事件。你再比了一场，但你输了。如果你今天赢了一场比赛，但这并不表示你明天肯定会赢。我们来分配一个随机变量X，用于表示赢得的次数。 X可能的值是多少呢？它可以是任意值，这取决于你掷硬币的次数。

只有两种可能的结果，成功和失败。因此，成功的概率 = 0.5，失败的概率可以很容易地计算得到：q = p – 1 = 0.5。

二项式分布就是只有两个可能结果的分布，比如成功或失败、得到或者丢失、赢或败，每一次尝试成功和失败的概率相等。

结果有可能不一定相等。如果在实验中成功的概率为0.2，则失败的概率可以很容易地计算得到 q = 1 - 0.2 = 0.8。

每一次尝试都是独立的，因为前一次投掷的结果不能决定或影响当前投掷的结果。只有两个可能的结果并且重复n次的实验叫做二项式。二项分布的参数是n和p，其中n是试验的总数，p是每次试验成功的概率。

在上述说明的基础上，二项式分布的属性包括：

公式

𝑁⋅𝑝 表示分布的均值

将left修改为x：

# IMPORTS
import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import matplotlib.style as style
from IPython.core.display import HTML

# PLOTTING CONFIG
%matplotlib inline
style.use('fivethirtyeight')
plt.rcParams["figure.figsize"] = (14, 7)

plt.figure(dpi=100)

# PDF
plt.bar(x=np.arange(20), 
        height=(stats.binom.pmf(np.arange(20), p=.5, n=20)), 
        width=.75,
        alpha=0.75
       )
# CDF
plt.plot(np.arange(20),
         stats.binom.cdf(np.arange(20), p=.5, n=20),
         color="#fc4f30",
        )

# LEGEND
plt.text(x=4.5, y=.7, s="pmf (normed)", alpha=.75, weight="bold", color="#008fd5")
plt.text(x=14.5, y=.9, s="cdf", alpha=.75, weight="bold", color="#fc4f30")

# TICKS
plt.xticks(range(21)[::2])
plt.tick_params(axis = 'both', which = 'major', labelsize = 18)
plt.axhline(y = 0.005, color = 'black', linewidth = 1.3, alpha = .7)

# TITLE, SUBTITLE & FOOTER
plt.text(x = -2.5, y = 1.25, s = "Binomial Distribution - Overview",
               fontsize = 26, weight = 'bold', alpha = .75)
plt.text(x = -2.5, y = 1.1, 
         s = 'Depicted below are the normed probability mass function (pmf) and the cumulative density\nfunction (cdf) of a Binomial distributed random variable $ y \sim Binom(N, p) $, given $ N = 20$ and $p =0.5 $.',
         fontsize = 19, alpha = .85)

改变P值后，对结果的影响如下图所示：

plt.figure(dpi=100)

# PDF P = .2
plt.scatter(np.arange(21),
            (stats.binom.pmf(np.arange(21), p=.2, n=20)),
            alpha=0.75,
            s=100
       )
plt.plot(np.arange(21),
         (stats.binom.pmf(np.arange(21), p=.2, n=20)),
         alpha=0.75,
        )

# PDF P = .5
plt.scatter(np.arange(21),
            (stats.binom.pmf(np.arange(21), p=.5, n=20)),
            alpha=0.75,
            s=100
       )
plt.plot(np.arange(21),
         (stats.binom.pmf(np.arange(21), p=.5, n=20)),
         alpha=0.75,
        )

# PDF P = .9
plt.scatter(np.arange(21),
            (stats.binom.pmf(np.arange(21), p=.9, n=20)),
            alpha=0.75,
            s=100
       )
plt.plot(np.arange(21),
         (stats.binom.pmf(np.arange(21), p=.9, n=20)),
         alpha=0.75,
        )

# LEGEND
plt.text(x=3.5, y=.075, s="$p = 0.2$", alpha=.75, weight="bold", color="#008fd5")
plt.text(x=9.5, y=.075, s="$p = 0.5$", alpha=.75, weight="bold", color="#fc4f30")
plt.text(x=17.5, y=.075, s="$p = 0.9$", alpha=.75, weight="bold", color="#e5ae38")

# TICKS
plt.xticks(range(21)[::2])
plt.tick_params(axis = 'both', which = 'major', labelsize = 18)
plt.axhline(y = 0, color = 'black', linewidth = 1.3, alpha = .7)

# TITLE, SUBTITLE & FOOTER
plt.text(x = -2.5, y = .37, s = "Binomial Distribution - $p$",
               fontsize = 26, weight = 'bold', alpha = .75)
plt.text(x = -2.5, y = .32, 
         s = 'Depicted below are three Binomial distributed random variables with varying $p $. As one can see\nthe parameter $p$ shifts and skews the distribution.',
         fontsize = 19, alpha = .85)

改变N值对结果产生的影响如下所示：

plt.figure(dpi=100)

# PDF N = 10
plt.scatter(np.arange(11),
            (stats.binom.pmf(np.arange(11), p=.5, n=10)),
            alpha=0.75,
            s=100
       )
plt.plot(np.arange(11),
         (stats.binom.pmf(np.arange(11), p=.5, n=10)),
         alpha=0.75,
        )

# PDF N = 15
plt.scatter(np.arange(16),
            (stats.binom.pmf(np.arange(16), p=.5, n=15)),
            alpha=0.75,
            s=100
       )
plt.plot(np.arange(16),
         (stats.binom.pmf(np.arange(16), p=.5, n=15)),
         alpha=0.75,
        )

# PDF N = 20
plt.scatter(np.arange(21),
            (stats.binom.pmf(np.arange(21), p=.5, n=20)),
            alpha=0.75,
            s=100
       )
plt.plot(np.arange(21),
         (stats.binom.pmf(np.arange(21), p=.5, n=20)),
         alpha=0.75,
        )

# LEGEND
plt.text(x=6, y=.225, s="$N = 10$", alpha=.75, weight="bold", color="#008fd5")
plt.text(x=8.5, y=.2, s="$N = 15$", alpha=.75, weight="bold", color="#fc4f30")
plt.text(x=11, y=.175, s="$N = 20$", alpha=.75, weight="bold", color="#e5ae38")

# TICKS
plt.xticks(range(21)[::2])
plt.tick_params(axis = 'both', which = 'major', labelsize = 18)
plt.axhline(y = 0, color = 'black', linewidth = 1.3, alpha = .7)

# TITLE, SUBTITLE & FOOTER
plt.text(x = -2.5, y = .31, s = "Binomial Distribution - $N$",
               fontsize = 26, weight = 'bold', alpha = .75)
plt.text(x = -2.5, y = .27, 
         s = 'Depicted below are three Binomial distributed random variables with varying $N$. As one can see\nthe parameter $N$ streches the distribution (the larger $N$ the flatter the distribution).',
         fontsize = 19, alpha = .85)

也可以构造随机样本（Random Variates）：

import numpy as np
from scipy.stats import binom

# draw a single sample
np.random.seed(42)
print(binom.rvs(p=0.3, n=10), end="\n\n")

# draw 10 samples
print(binom.rvs(p=0.3, n=10, size=10), end="\n\n")

计算概率概率质量函数的概率：

from scipy.stats import binom

# additional imports for plotting purpose
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.rcParams["figure.figsize"] = (14,7)

# likelihood of x and y
x = 1
y = 7
print("pmf(X=1) = {}\npmf(X=7) = {}".format(binom.pmf(k=x, p=0.3, n=10), binom.pmf(k=y, p=0.3, n=10)))

# continuous pdf for the plot
x_s = np.arange(11)
y_s = binom.pmf(k=x_s, p=0.3, n=10)
plt.scatter(x_s, y_s, s=100);

计算累积概率密度函数的概率：

from scipy.stats import binom

# probability of x less or equal 0.3
print("P(X <=3) = {}".format(binom.cdf(k=3, p=0.3, n=10)))

# probability of x in [-0.2, +0.2]
print("P(2 < X <= 8) = {}".format(binom.cdf(k=8, p=0.3, n=10) - binom.cdf(k=2, p=0.3, n=10)))