总体 (任意分布) → 抽 n 个 → 算样本均值 X̄ → X̄ 也有分布

import numpy as np
import matplotlib.pyplot as plt

# 总体: 严重右偏的指数分布 (非正态)
np.random.seed(42)
population = np.random.exponential(scale=2, size=100_000)

plt.figure(figsize=(12, 4))
plt.subplot(1, 3, 1)
plt.hist(population, bins=50, density=True, color="lightcoral")
plt.title(f"总体: 指数分布 (右偏)\nμ={population.mean():.2f}, σ={population.std():.2f}")

# 抽不同 n, 看 X̄ 的分布
for i, n in enumerate([1, 5, 30, 100], start=2):
    sample_means = [np.mean(np.random.choice(population, n)) for _ in range(10_000)]
    plt.subplot(1, 5, i)
    plt.hist(sample_means, bins=30, density=True, color="steelblue", alpha=0.7)
    plt.title(f"n={n}\nμ̄={np.mean(sample_means):.2f}, σ̄={np.std(sample_means):.2f}")
plt.tight_layout(); plt.show()

# 经验: 增加样本的"性价比"
import numpy as np
import matplotlib.pyplot as plt
n_seq = np.arange(10, 10001)
se = 10 / np.sqrt(n_seq)
plt.plot(n_seq, se)
plt.xlabel("样本量 n"); plt.ylabel("标准误 σ/√n")
plt.title("样本量 vs 标准误 (σ=10)")
plt.grid(True); plt.show()

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

x = np.linspace(-4, 4, 200)
plt.plot(x, norm.pdf(x), "k-", label="N(0,1)", linewidth=2)
for df in [1, 5, 30]:
    plt.plot(x, t.pdf(x, df), "--", label=f"t({df})")
plt.legend(); plt.title("t 分布 vs 正态 (df 越小, 尾部越厚)")
plt.show()

# 演示 LLN
import numpy as np
import matplotlib.pyplot as plt

np.random.seed(42)
# 抛硬币, 看累积频率
flips = np.random.choice([0, 1], 10000)
cum_mean = np.cumsum(flips) / np.arange(1, 10001)
plt.plot(cum_mean)
plt.axhline(0.5, color="red", linestyle="--", label="理论 0.5")
plt.xlabel("抛硬币次数"); plt.ylabel("累积频率")
plt.title("大数定律: 累积频率 → 0.5")
plt.legend(); plt.grid(True); plt.show()

# 例: 中位数置信区间 (中位数不是正态)
import numpy as np
from scipy.stats import bootstrap

# 偏态数据: 收入
np.random.seed(42)
incomes = np.random.lognormal(10.5, 0.5, 200)  # 200 个高收入样本

# 中位数及其 95% 置信区间
median = np.median(incomes)
boot = bootstrap((incomes,), np.median, n_resamples=10000,
                 confidence_level=0.95, random_state=42)
ci_low, ci_high = boot.confidence_interval
print(f"中位数: {median:.0f}, 95% CI: [{ci_low:.0f}, {ci_high:.0f}]")

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

# 设定: 某市成年男性身高 N(173, 8²), 抽 30 个
mu, sigma = 173, 8
n = 30
sample_size = 10000  # 模拟 10000 次抽样

# 1. 样本均值的均值和标准差 (理论: μ, σ/√n)
sample_means = np.random.normal(mu, sigma, (sample_size, n)).mean(axis=1)
print(f"X̄ 的均值: {sample_means.mean():.2f} (理论 {mu})")
print(f"X̄ 的标准差: {sample_means.std():.3f} (理论 {sigma/np.sqrt(n):.3f})")

# 2. 画图: X̄ 分布 + 理论正态
plt.figure(figsize=(8, 5))
plt.hist(sample_means, bins=40, density=True, alpha=0.6, label="模拟")
x = np.linspace(mu - 3, mu + 3, 200)
plt.plot(x, norm.pdf(x, mu, sigma/np.sqrt(n)), "r-",
         label=f"N({mu}, {sigma/np.sqrt(n):.2f}²)")
plt.xlabel("样本均值 X̄"); plt.ylabel("密度")
plt.title(f"n={n} 时, X̄ 的抽样分布 vs 理论正态")
plt.legend(); plt.show()

# 3. 用 t 分布找 95% 置信区间 (假设 σ 未知)
# 实际抽 1 个样本
sample = np.random.normal(mu, sigma, n)
xbar = sample.mean()
s = sample.std(ddof=1)
t_crit = t.ppf(0.975, df=n-1)  # 95% 置信度
ci_low = xbar - t_crit * s / np.sqrt(n)
ci_high = xbar + t_crit * s / np.sqrt(n)
print(f"\n1 个样本均值: {xbar:.2f}, 95% CI: [{ci_low:.2f}, {ci_high:.2f}]")