import numpy as np
import matplotlib.pyplot as pltX = np.random.rand(1000)fig, (axs1, axs2) = plt.subplots(1, 2, figsize=(9, 4))axs1.plot(X, '.')
axs1.set_ylim(-2, 2)axs2.hist(X, edgecolor='k', alpha=0.8)
plt.show()print("mean:", np.round(X.mean(), 2), "std:", np.round(X.std(), 2))

mean: 0.5 std: 0.29

W = (X - X.mean())/X.std()mean = W.mean()
std_var = W.std()fig, (axs1, axs2) = plt.subplots(1, 2, figsize=(9, 4))axs1.plot(W, '.')
axs1.set_ylim(-2, 2)axs2.hist(W, edgecolor='k', alpha=0.8)
plt.show()print("mean:", np.round(W.mean(), 2), "std:", np.round(W.std(), 2))

mean: -0.0 std: 1.0

def NormDist(x, mean=0, sigma=1):assert(sigma != 0)return np.exp(-(x-mean)**2/2)/(np.sqrt(2*np.pi)*sigma)plt.figure(figsize=(8, 6))xs = np.linspace(-5, 5, 100)
X = [NormDist(i) for i in xs]
plt.plot(xs, X, '-', label=r'$frac{1}{sqrt{2pi}sigma}exp(frac{-x^2}{2})$')X = [np.exp(-i**2/2) for i in xs]
plt.plot(xs, X, '--', label=r'$exp(frac{-x^2}{2})$')X = [np.exp(-i**2) for i in xs]
plt.plot(xs, X, '--', label=r'$exp(-x^2)$')plt.legend(fontsize=14)
plt.show()

sample_size = 1000 # 采样个数
number_of_sample = 1 # 采样次数np.random.seed(0)
X = np.random.rand(number_of_sample, sample_size)mean = X.mean(axis=1)fig, axs = plt.subplots(1, 2, figsize=(9, 4))# sample values
axs[0].plot(X[0], '.')
axs[0].set_ylim(-1, 2)
axs[0].set_title('sample values')# histgram of X
axs[1].hist(X[0], edgecolor='k', alpha=0.8)
axs[1].set_title('histgram of $X$')plt.show()
print('mean:', mean)

mean: [0.49592153]

from scipy.stats import skewnorm
import matplotlib.pyplot as plta = 4
fig, ax = plt.subplots(1, 1)
mean, var, skew, kurt = skewnorm.stats(a, moments='mvsk')x = np.linspace(skewnorm.ppf(0.01, a), skewnorm.ppf(0.99, a), 100)
ax.plot(x, skewnorm.pdf(x, a), 'r-', lw=5, alpha=0.6, label='skewnorm pdf')print('E(X) =', mean)E(X) = 0.7740617226446519

r = skewnorm.rvs(a, size=1000)fig, axs = plt.subplots(1, 2, figsize=(9, 4))# sample values
axs[0].plot(r, '.')
axs[0].set_ylim(-2, 5)
axs[0].set_title('sample values')# histgram of X
axs[1].hist(r, edgecolor='k', alpha=0.8)
axs[1].set_title('histgram of $X$')print('mean:', r.mean())mean: 0.7426786528215207

sample_size = [3, 10, 30, 100] # 每次采样个数
number_of_sample = 100000 # 采样次数plt.figure(figsize=(8,6))for i, ss in enumerate(sample_size):Xs = np.random.rand(number_of_sample, ss)    h, b = np.histogram(Xs.mean(axis=1), bins=50, density=True)b = (b[:-1] + b[1:]) / 2.plt.plot(b, h, '-', label=f'sample size = {ss}')plt.legend()

sample_size = [3, 10, 30, 100] # 每次采样个数
number_of_sample = 100000 # 采样次数plt.figure(figsize=(8,6))for i, ss in enumerate(sample_size):Xs = skewnorm.rvs(4, size=(number_of_sample, ss))h, b = np.histogram(Xs.mean(axis=1), bins=50, density=True)b = (b[:-1] + b[1:]) / 2.plt.plot(b, h, '-', label=f'sample size = {ss}')plt.legend()