Bernoulli Distribution with Python from Scratch

To solve a problem, the common ready libraries are used but we don’t search the function how to work and how to serve our purpose. More importantly, if we need to modify a function, we don’t know how to do that.

So, the purpose of this article is how to code bernoulli Probability distributions and their some properties in a simple way, without using ready libraries like "SciPy" and gain some basic skills from scratch.

The bernoulli distribution is a discrete distribution that is used when a random experiment is performed and only two results are obtained such as good-bad, positive-negative, success-failure. Those statements are used to describe the probabilities of an event. Bernoulli trial is the simple way to represent an experiment like the outcome of a coin heads or tails, the result of an exam pass or failure, etc.

Probability Function

If X is a random variable and p is a probability of an event with this distribution, then:

Python Code:

#bernoulli probability mass function:

def pmf(x,p):
    f = p**x*(1-p)**(1-x)
    return f

Mean

The mean or expected value E(x) of Bernoulli random variable X is:

Python Code:

# expected value of x
def mean(p):
    return p

Variance and Standart Deviation

The variance of random variable X is:

Python Code:

#variance of x
def var(p):
    return p*(1-p)

#standard deviation is root of variance
def std(p):
    return var(p)**(1/2)

Generate Random Variates

To generate random variates corresponding to Bernoulli distribution

Python Code

import numpy as np

#size is a parameter that how many number generates
def rvs(p,size=1):
    rvs = np.array([])
    for i in range(0,size):
        if np.random.rand() <= p:
            a=1
            rvs = np.append(rvs,a)
        else:
            a=0
            rvs = np.append(rvs,a)
    return rvs

Let put them together

import numpy as np

#created a bernoulli class

class bernoulli():

    def pmf(x,p):
        """
        probability mass function        
        """
        f = p**x*(1-p)**(1-x)
        return f

    def mean(p):
        """
        expected value of bernoulli random variable
        """
        return p

    def var(p):
        """
        variance of bernoulli random variable
        """
        return p*(1-p)

    def std(p):
        """
        standart deviation of bernoulli random variable
        """
        return bernoulli.var(p)**(1/2)

    def rvs(p,size=1):
        """
        random variates
        """
        rvs = np.array([])
        for i in range(0,size):
            if np.random.rand() <= p:
                a=1
                rvs = np.append(rvs,a)
            else:
                a=0
                rvs = np.append(rvs,a)
        return rvs

Example

Suppose that a formula-1 racer has a 0.2 probability of having an accident. So that X = 1 if an accident occurs, and X = 0 otherwise. Find the expectation, variance and standart deviation of X and generate random variate 10 times.

p=0.2 # probability of having an accident

bernoulli.mean(p) # return -> 0.2
bernoulli.var(p) # return -> 0.16
bernoulli.std(p) # return -> 0.4

#each execution generates random numbers, so array may be change
bernoulli.rvs(p,size=10) 
#return-> array([0., 0., 0., 0., 1., 0., 1., 0., 0., 1.])

I hope this article is useful and gives you a different perspective.

References:

Bernoulli Distribution https://www.wikiwand.com/en/Bernoulli_distribution

Probability Distribution https://www.wikiwand.com/en/Probability_distribution