How to generate random numbers from a log-normal distribution in Python ?

Last Updated : 07 Apr, 2021

A continuous probability distribution of a random variable whose logarithm is usually distributed is known as a log-normal (or lognormal) distribution in probability theory.

A variable x is said to follow a log-normal distribution if and only if the log(x) follows a normal distribution. The PDF is defined as follows.

Probability Density function Log-normal

Where mu is the population mean & sigma is the standard deviation of the log-normal distribution of a variable. Just like normal distribution which is a manifestation of summation of a large number of Independent and identically distributed random variables, lognormal is the result of multiplying a large number of Independent and identically distributed random variables. Generating a random number from a log-normal distribution is very easy with help of the NumPy library.

Syntax:

numpy.random.lognormal(mean=0.0, sigma=1.0, size=None)

Parameter:

mean: It takes the mean value for the underlying normal distribution.

sigma: It takes only non-negative values for the standard deviation for the underlying normal distribution

size : It takes either a int or a tuple of given shape. If a single value is passed it returns a single integer as result. If a tuple then it returns a 2D matrix of values from log-normal distribution.

Returns: Drawn samples from the parameterized log-normal distribution(nd Array or a scalar).

The below example depicts how to generate random numbers from a log-normal distribution:

Python3

# import modules 
import numpy as np 
import matplotlib.pyplot as plt 
  
# mean and standard deviation 
mu, sigma = 3., 1.  
s = np.random.lognormal(mu, sigma, 10000) 
  
# depict illustration 
count, bins, ignored = plt.hist(s, 30, 
                                density=True,  
                                color='green') 
x = np.linspace(min(bins), 
                max(bins), 10000) 
  
pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) 
       / (x * sigma * np.sqrt(2 * np.pi))) 
  
# assign other attributes 
plt.plot(x, pdf, color='black') 
plt.grid() 
plt.show() 

Output:

Let’s prove that log-Normal is a product of independent and identical distributions of a random variable using python. In the program below we are generating 1000 points randomly from a normal distribution and then taking the product of them and finally plotting it to get a log-normal distribution.

Python3

# Importing required modules 
import numpy as np 
import matplotlib.pyplot as plt 
  
b = [] 
  
# Generating 1000 points from normal distribution. 
for i in range(1000): 
    a = 12. + np.random.standard_normal(100) 
    b.append(np.product(a)) 
  
# Making all negative values  into positives 
b = np.array(b) / np.min(b) 
count, bins, ignored = plt.hist(b, 100,  
                                density=True,  
                                color='green') 
  
sigma = np.std(np.log(b)) 
mu = np.mean(np.log(b)) 
  
# Plotting the graph. 
x = np.linspace(min(bins), max(bins), 10000) 
pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) 
       / (x * sigma * np.sqrt(2 * np.pi))) 
  
plt.plot(x, pdf,color='black') 
plt.grid() 
plt.show() 