Python – Binomial Distribution
Binomial distribution
Bernoulli
- 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.
r
n
r
n-r
r
probability mass function(pmf)
r
n-r
6
0.6
success’
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 |
np
np(1-p)
Using Python to obtain the distribution :
SciPy
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
scipy.stats
stats()
scipy.stats.binom
n
p
Syntax : scipy.stats.binom.stats(n, p)
scipy.stats.binom.pmf()
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 p n = 6p = 0.6# defining the list of r values r_values = list(range(n + 1)) # obtaining the mean and variance mean, var = binom.stats(n, p) # list of pmf values dist = [binom.pmf(r, n, p) for r in r_values ] # printing the table print("r\tp(r)") for i in range(n + 1): print(str(r_values[i]) + "\t" + str(dist[i])) # printing mean and variance print("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 binom import matplotlib.pyplot as plt # setting the values # of n and p n = 6p = 0.6# defining list of r values r_values = list(range(n + 1)) # list of pmf values dist = [binom.pmf(r, n, p) for r in r_values ] # plotting the graph plt.bar(r_values, dist) plt.show() |
Output :
normal
p
0.5