이산확률분포 그려보기 (Python)

파이썬 코드로 이산확률분포가 파라미터에 따라 어떻게 그려지는지 알아보자.

import numpy as np
from scipy.special import binom, comb
import matplotlib.pyplot as plt

Binomial Distribution, 이항분포

def binomial(x, n, theta):
    return comb(n, x) * (theta ** x) * ((1 - theta) ** (n - x))
    
params = [(20, 0.3), (20, 0.5), (20, 0.7), (26, 0.9)]
fig, axes = plt.subplots(1, 4, figsize=(24, 4))
for ax, (n, theta) in zip(axes, params):
    print(n, theta)
    probs = []
    for x in range(n + 1):
        probs.append(binomial(x, n, theta))
    ax.bar(range(n + 1), probs)
    ax.set_title(f'n={n}, theta={theta}')

Figure 1. Binomial distribution

Geometric distribution, 기하분포

기하분포는 단조감소하는 형태이다.

def geometric(x, theta):
    return ((1 - theta) ** x) * theta
    
params = [(10, 0.3), (10, 0.5), (10, 0.7), (10, 0.9)]
fig, axes = plt.subplots(1, 4, figsize=(26, 4))
for ax, (n, theta) in zip(axes, params):
    probs = []
    for x in range(n + 1):
        probs.append(geometric(x, theta))
    ax.bar(range(n + 1), probs)
    ax.set_title(f'theta={theta}')
    ax.set_xlabel('X=x')
    ax.set_ylabel('Probability')

Figure 2. Geometric distribution

 

Negative-Binomial distribution, 음이항분포

def neg_bin(x, r, theta):
    return comb(r - 1 + x, x) * (theta ** r) * ((1 - theta) ** x)
    
params = [(10, 2, 0.6), (12, 4, 0.6), (20, 6, 0.6), (20, 8, 0.6)]
fig, axes = plt.subplots(1, 4, figsize=(16, 4))
for ax, (n, r, theta) in zip(axes, params):
    probs = []
    for x in range(n + 1):
        probs.append(neg_bin(x, r, theta))
    ax.bar(range(n + 1), probs)
    ax.set_title(f'r={r}, theta={theta}')
    ax.set_xlabel('X=x')
    ax.set_ylabel('Probability')

Figure 3. Negative-Binomal distribution

 

Poisson distribution, 포아송분포

포아송분포는 단 하나의 peak가 존재한다.

포아송은 수식의 시작이 이항분포이므로 역시 이항분포와 비슷한 분포를 갖는다.

def poisson(x, lambda_):
    return ((lambda_ ** x) * np.math.exp(-lambda_)) / np.math.factorial(x)

lambdas = [1, 3, 5, 7]
fig, axes = plt.subplots(1, 4, figsize=(26, 4))
for ax, (lambda_) in zip(axes, lambdas):
    probs, n = [], lambda_ * 3
    for x in range(n + 1):
        probs.append(poisson(x, lambda_))
    ax.bar(range(n + 1), probs)
    ax.set_title(f'lambda={lambda_}')
    ax.set_xlabel('X=x')
    ax.set_ylabel('Probability')

Figure 4. Poisson distribution

 

Hypergeometric distribution, 초기하분포

초기하분포는 하나의 peak가 존재한다

그리고 $N$과 $M$이 충분히 크면 이항분포에 근사하는 것도 확인할 수 있다.

def hyper_geo(x, n, N, M):
    return (comb(M, x) * comb(N - M, n - x)) / comb(N, n)

params = [(5, 15, 5), (5, 15, 7), (50, 100, 80), (300, 500, 70)] # (n, N, M)
fig, axes = plt.subplots(1, 4, figsize=(26, 4))
for ax, (n, N, M) in zip(axes, params):
    probs = []
    x_min = max(0, n + M - N)
    x_max = min(n, M)
    for x in range(x_min, x_max + 1):
        probs.append(hyper_geo(x, n, N, M))
    ax.bar(range(x_min, x_max + 1), probs)
    ax.set_title(f'n={n}, N={N}, M={M}')
    ax.set_xlabel('X=x')
    ax.set_ylabel('Probability')

Figure 5. Hypergeometric distribution