import matplotlib.pyplot as plt
import numpy as np
xs=np.linspace(-1,1, num=1000)
from scipy.stats import norm
sigma=1
n=16
cdf_16 = norm.pdf(xs, loc=0, scale=sigma/np.sqrt(n))
plt.plot(xs, cdf_16, label=f"n ={n}")
n=32
cdf_32 = norm.pdf(xs, loc=0, scale=sigma/np.sqrt(n))
plt.plot(xs, cdf_32, label=f"n ={n}")
n=64
cdf_64 = norm.pdf(xs, loc=0, scale=sigma/np.sqrt(n))
plt.plot(xs, cdf_64, label=f"n ={n}")

plt.axvline(0, label=r"$\mu=0$", color="red")
plt.ylabel("pdf")
plt.legend()
plt.xlabel(r"$\overline{X}_{n}$")
plt.title(r"Sampling distribution of $\overline{X}_{n}=\frac{1}{n} \sum_{i=1}^n X_i$, with $X_{i}\sim \mathcal{N}(\mu=0, \sigma = 1)$");