正态分布（Normal distribution），也称“常态分布”，又名高斯分布（Gaussian distribution）

正态曲线呈钟型，两头低，中间高，左右对称因其曲线呈钟形，因此人们又经常称之为钟形曲线。

若随机变量X服从一个数学期望为μ、方差为σ^2的正态分布。其概率密度函数为正态分布的期望值μ决定了其位置，其标准差σ决定了分布的幅度。当μ = 0,σ = 1时的正态分布是标准正态分布。

用python 模拟

#!/usr/bin/python# -*- coding:utf-8 -*-import numpy as npfrom scipy import statsimport mathimport matplotlib as mplimport matplotlib.pyplot as pltfrom mpl_toolkits.mplot3d import Axes3Dfrom matplotlib import cmimport seaborndef calc_statistics(x): n = x.shape[0] # 样本个数 # 手动计算 m = 0 m2 = 0 m3 = 0 m4 = 0 for t in x: m += t m2 += t*t m3 += t**3 m4 += t**4 m /= n m2 /= n m3 /= n m4 /= n mu = m sigma = np.sqrt(m2 - mu*mu) skew = (m3 - 3*mu*m2 + 2*mu**3) / sigma**3 kurtosis = (m4 - 4*mu*m3 + 6*mu*mu*m2 - 4*mu**3*mu + mu**4) / sigma**4 - 3 print('手动计算均值、标准差、偏度、峰度：', mu, sigma, skew, kurtosis) # 使用系统函数验证 mu = np.mean(x, axis=0) sigma = np.std(x, axis=0) skew = stats.skew(x) kurtosis = stats.kurtosis(x) return mu, sigma, skew, kurtosisif __name__ == '__main__': d = np.random.randn(10000) print(d) print(d.shape) mu, sigma, skew, kurtosis = calc_statistics(d) print('函数库计算均值、标准差、偏度、峰度：', mu, sigma, skew, kurtosis) # 一维直方图 mpl.rcParams['font.sans-serif'] = 'SimHei' mpl.rcParams['axes.unicode_minus'] = False plt.figure(num=1, facecolor='w') y1, x1, dummy = plt.hist(d, bins=30, normed=True, color='g', alpha=0.75, edgecolor='k', lw=0.5) t = np.arange(x1.min(), x1.max(), 0.05) y = np.exp(-t**2 / 2) / math.sqrt(2*math.pi) plt.plot(t, y, 'r-', lw=2) plt.title('高斯分布，样本个数：%d' % d.shape[0]) plt.grid(b=True, ls=':', color='#404040') # plt.show() d = np.random.randn(100000, 2) mu, sigma, skew, kurtosis = calc_statistics(d) print('函数库计算均值、标准差、偏度、峰度：', mu, sigma, skew, kurtosis) # 二维图像 N = 30 density, edges = np.histogramdd(d, bins=[N, N]) print('样本总数：', np.sum(density)) density /= density.max() x = y = np.arange(N) print('x = ', x) print('y = ', y) t = np.meshgrid(x, y) print(t) fig = plt.figure(facecolor='w') ax = fig.add_subplot(111, projection='3d') # ax.scatter(t[0], t[1], density, c='r', s=50*density, marker='o', depthshade=True, edgecolor='k') ax.plot_surface(t[0], t[1], density, cmap=cm.Accent, rstride=1, cstride=1, alpha=0.9, lw=0.75, edgecolor='k') ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_zlabel('Z') plt.title('二元高斯分布，样本个数：%d' % d.shape[0], fontsize=15) plt.tight_layout(0.1) plt.show()

来个6的

二元高斯分布方差比较

#!/usr/bin/python# -*- coding:utf-8 -*-import numpy as npfrom scipy import statsimport matplotlib as mplimport matplotlib.pyplot as pltfrom mpl_toolkits.mplot3d import Axes3Dfrom matplotlib import cmif __name__ == '__main__': x1, x2 = np.mgrid[-5:5:51j, -5:5:51j] x = np.stack((x1, x2), axis=2) print('x1 = \n', x1) print('x2 = \n', x2) print('x = \n', x) mpl.rcParams['axes.unicode_minus'] = False mpl.rcParams['font.sans-serif'] = 'SimHei' plt.figure(figsize=(9, 8), facecolor='w') sigma = (np.identity(2), np.diag((3,3)), np.diag((2,5)), np.array(((2,1), (1,5)))) for i in np.arange(4): ax = plt.subplot(2, 2, i+1, projection='3d') norm = stats.multivariate_normal((0, 0), sigma[i]) y = norm.pdf(x) ax.plot_surface(x1, x2, y, cmap=cm.Accent, rstride=1, cstride=1, alpha=0.9, lw=0.3, edgecolor='#303030') ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_zlabel('Z') plt.suptitle('二元高斯分布方差比较', fontsize=18) plt.tight_layout(1.5) plt.show()

图像好看吗？

以上这篇使用python模拟高斯分布例子就是小编分享给大家的全部内容了，希望能给大家一个参考，也希望大家多多支持。