您要求的是预期直方图的图表 .

假设[a，b]是直方图的x个区间之一 . 对于大小为n的随机样本，间隔中的预期样本数为

(cdf(b) - cdf(a))*n

其中cdf(x)是累积分布函数 . 要绘制预期的直方图，您将计算每个bin的值 .

下面的脚本显示了在matplotlib直方图上绘制预期直方图的一种方法 . 它生成这个图：

import numpy as np

import scipy.stats

import matplotlib.pyplot as plt

# Generate log-normal distributed set of samples

np.random.seed(1234)

samples = np.random.lognormal(mean=1., sigma=.4, size=10000)

# Make a fit to the samples.

shape, loc, scale = scipy.stats.lognorm.fit(samples, floc=0)

# Create the histogram plot using matplotlib. The first two values in

# the tuple returned by hist are the number of samples in each bin and

# the values of the histogram's bin edges. counts has length num_bins,

# and edges has length num_bins + 1.

num_bins = 50

clr = '#FFE090'

counts, edges, patches = plt.hist(samples, bins=num_bins, color=clr, label='Sample histogram')

# Create an array of length num_bins containing the center of each bin.

centers = 0.5*(edges[:-1] + edges[1:])

# Compute the CDF at the edges. Then prob, the array of differences,

# is the probability of a sample being in the corresponding bin.

cdf = scipy.stats.lognorm.cdf(edges, shape, loc=loc, scale=scale)

prob = np.diff(cdf)

plt.plot(centers, samples.size*prob, 'k-', linewidth=2, label='Expected histogram')

# prob can also be approximated using the PDF at the centers multiplied

# by the width of the bin:

# p = scipy.stats.lognorm.pdf(centers, shape, loc=loc, scale=scale)

# prob = p*(edges[1] - edges[0])

# plt.plot(centers, samples.size*prob, 'r')

plt.legend()

plt.show()

注意：由于PDF是CDF的衍生物，您可以将cdf(b) - cdf(a)的近似值写为

cdf(b) - cdf(a) = pdf(m)*(b - a)

其中m是，例如，区间[a，b]的中点 . 然后，您提出的确切问题的答案是通过将PDF乘以样本大小和直方图区域宽度来缩放PDF . 脚本中有一些注释掉的代码，显示如何使用缩放的PDF绘制预期的直方图 . 但由于CDF也可用于对数正态分布，因此您也可以使用它 .