Binomial distribution is a probability distribution that summarises the likelihood that a variable will take one of two independent values under a given set of parameters. The distribution is obtained by performing a number of Bernoulli trials.
A Bernoulli trial is assumed to meet each of these criteria :
- There must be only 2 possible outcomes.
- Each outcome has a fixed probability of occurring. A success has the probability of p, and a failure has the probability of 1 – p.
- Each trial is completely independent of all others.
The binomial random variable represents the number of successes(r) in n successive independent trials of a Bernoulli experiment.
Probability of achieving r success and n-r failure is :
The number of ways we can achieve r successes is : 
Hence, the probability mass function(pmf), which is the total probability of achieving r success and n-r failure is :
An example illustrating the distribution :
Consider a random experiment of tossing a biased coin 6 times where the probability of getting a head is 0.6. If ‘getting a head’ is considered as ‘success’ then, the binomial distribution table will contain the probability of r successes for each possible value of r.
| r | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| P(r) | 0.004096 | 0.036864 | 0.138240 | 0.276480 | 0.311040 | 0.186624 | 0.046656 |
This distribution has a mean equal to np and a variance of np(1-p).
Using Python to obtain the distribution :
Now, we will use Python to analyse the distribution(using SciPy) and plot the graph(using Matplotlib).
Modules required :
- SciPy:
SciPy is an Open Source Python library, used in mathematics, engineering, scientific and technical computing.Installation :
pip install scipy
- Matplotlib:
Matplotlib is a comprehensive Python library for plotting static and interactive graphs and visualisations.Installation :
pip install matplotlib
The scipy.stats module contains various functions for statistical calculations and tests. The stats() function of the scipy.stats.binom module can be used to calculate a binomial distribution using the values of n and p.
Syntax : scipy.stats.binom.stats(n, p)
It returns a tuple containing the mean and variance of the distribution in that order.
scipy.stats.binom.pmf() function is used to obtain the probability mass function for a certain value of r, n and p. We can obtain the distribution by passing all possible values of r(0 to n).
Syntax : scipy.stats.binom.pmf(r, n, p)
Calculating distribution table :
Approach :
- Define n and p.
- Define a list of values of r from 0 to n.
- Get mean and variance.
- For each r, calculate the pmf and store in a list.
Code :
from scipy.stats import binom# setting the values# of n and pn = 6p = 0.6# defining the list of r valuesr_values = list(range(n + 1))# obtaining the mean and variance mean, var = binom.stats(n, p)# list of pmf valuesdist = [binom.pmf(r, n, p) for r in r_values ]# printing the tableprint("r\tp(r)")for i in range(n + 1): print(str(r_values[i]) + "\t" + str(dist[i]))# printing mean and varianceprint("mean = "+str(mean))print("variance = "+str(var)) |
Output :
r p(r) 0 0.004096000000000002 1 0.03686400000000005 2 0.13824000000000003 3 0.2764800000000001 4 0.31104 5 0.18662400000000007 6 0.04665599999999999 mean = 3.5999999999999996 variance = 1.44
Code: Plotting the graph using matplotlib.pyplot.bar() function to plot vertical bars.
from scipy.stats import binomimport matplotlib.pyplot as plt# setting the values# of n and pn = 6p = 0.6# defining list of r valuesr_values = list(range(n + 1))# list of pmf valuesdist = [binom.pmf(r, n, p) for r in r_values ]# plotting the graph plt.bar(r_values, dist)plt.show() |
Output :
When success and failure are equally likely, the binomial distribution is a normal distribution. Hence, changing the value of p to 0.5, we obtain this graph, which is identical to a normal distribution plot :
