import numpy as np
import matplotlib.pyplot as plt

# 例: 抛 2 个骰子的点数和分布 (理论)
sums = np.arange(2, 13)
probs = np.array([1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1]) / 36  # C(2, sum-1) ... 等

plt.figure(figsize=(8, 4))
plt.bar(sums, probs, color="steelblue", edgecolor="black")
plt.xticks(sums); plt.xlabel("两个骰子点数和"); plt.ylabel("概率")
plt.title("均匀骰子的点数和分布")
plt.show()

# 例: 抛 2 个骰子, 点数和的期望
from itertools import product
outcomes = list(product(range(1, 7), repeat=2))
sums = [a + b for a, b in outcomes]
probs = [1/36] * 36
expected_value = sum(s * p for s, p in zip(sums, probs))
print(f"E(两个骰子点数和) = {expected_value}")  # 7.0

import numpy as np
# 抛 2 骰子点数和的方差
sums_arr = np.array(sums)
var = ((sums_arr - expected_value)**2).mean()
print(f"Var = {var:.3f}, σ = {np.sqrt(var):.3f}")
# Var ≈ 5.83, σ ≈ 2.42

from scipy.stats import binom
import numpy as np

# 例: n=10, p=0.5, 抛 10 次硬币, 正面次数的分布
n, p = 10, 0.5
x = np.arange(0, n + 1)
pmf = binom.pmf(x, n, p)

print(f"E(X) = {n*p}, Var(X) = {n*p*(1-p)}")
print(f"P(X=5) = {pmf[5]:.3f}")  # 0.246
print(f"P(X≥6) = {1 - binom.cdf(5, n, p):.3f}")  # 0.377

# 假设: A 组 1000 人 120 转化, B 组 1000 人 150 转化
# B 组真的更好吗? 用二项分布置信区间
from scipy.stats import binom
n_a, x_a = 1000, 120
n_b, x_b = 1000, 150

# 用 Wilson 区间
def wilson_ci(x, n, alpha=0.05):
    from scipy.stats import norm
    z = norm.ppf(1 - alpha / 2)
    p_hat = x / n
    denom = 1 + z**2 / n
    center = (p_hat + z**2 / (2 * n)) / denom
    margin = z * np.sqrt(p_hat * (1 - p_hat) / n + z**2 / (4 * n**2)) / denom
    return center - margin, center + margin

ci_a = wilson_ci(x_a, n_a)
ci_b = wilson_ci(x_b, n_b)
print(f"A 转化率: {x_a/n_a:.1%}, 95% CI: [{ci_a[0]:.1%}, {ci_a[1]:.1%}]")
print(f"B 转化率: {x_b/n_b:.1%}, 95% CI: [{ci_b[0]:.1%}, {ci_b[1]:.1%}]")
# 如果 CI 不重叠 → B 显著更好

from scipy.stats import poisson
import numpy as np

# 例: 平均每小时 3 通电话, 问 1 小时内 ≥ 5 通电话的概率
lam = 3
p_5_or_more = 1 - poisson.cdf(4, lam)
print(f"P(X≥5) = {p_5_or_more:.3f}")
# 0.185

# 监控 API 每分钟请求数, 历史平均 100, 突然到 500 算异常?
from scipy.stats import poisson
lam = 100  # 历史平均
threshold_high = poisson.ppf(0.999, lam)  # 99.9% 分位数
print(f"正常范围上限: {threshold_high}")  # ≈ 149
# 500 远超 149, 几乎肯定是异常

from scipy.stats import geom
# 平均 1/0.1 = 10 次出一次正面, 问前 5 次都没正面的概率
print(geom.pmf(6, 0.1))  # 第 6 次才出现: (0.9)^5 × 0.1 = 0.059

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom, poisson, norm

# 1. 二项分布模拟
np.random.seed(42)
samples_binom = np.random.binomial(n=20, p=0.3, size=10000)
# 经验分布 vs 理论
plt.figure(figsize=(12, 4))
plt.subplot(1, 2, 1)
plt.hist(samples_binom, bins=21, density=True, alpha=0.7, label="模拟")
x = np.arange(0, 21)
plt.plot(x, binom.pmf(x, 20, 0.3), "ro-", label="理论")
plt.title("二项分布: n=20, p=0.3"); plt.legend()

# 2. 泊松分布模拟
samples_pois = np.random.poisson(lam=4, size=10000)
plt.subplot(1, 2, 2)
plt.hist(samples_pois, bins=20, density=True, alpha=0.7, label="模拟")
x = np.arange(0, 20)
plt.plot(x, poisson.pmf(x, 4), "ro-", label="理论")
plt.title("泊松分布: λ=4"); plt.legend()
plt.tight_layout(); plt.show()