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.

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:

# 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.

# 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()

Output: