Pdf and cdf in python

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Prerequisites: Matplotlib 

Matplotlib is a library in Python and it is a numerical — mathematical extension for the NumPy library.  The cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Properties of CDF:

• Every cumulative distribution function F(X) is non-decreasing
• If maximum value of the cdf function is at x, F(x) = 1.
• The CDF ranges from 0 to 1.

Method 1: Using the histogram

CDF can be calculated using PDF (Probability Distribution Function). Each point of random variable will contribute cumulatively to form CDF.

Example : 

A combination set containing 2 balls which can be either red or blue can be in the following set.

{RR, RB, BR, BB}

t -> No of red balls.

P(x = t) -> t = 0 : 1 / 4 [BB] 

            t = 1 : 2 / 4 [RB, BR]

            t = 2 : 1 / 4 [RR]

CDF :

F(x) = P(x<=t)

x = 0 : P(0)               -> 1 / 4

x = 1 : P(1) + P(0)        -> 3 / 4

x = 2 : P(2) + P(1) + P(0) -> 1

Approach

• Import modules
• Declare number of data points
• Initialize random values
• Plot histogram using above data
• Get histogram data
• Finding PDF using histogram data
• Calculate CDF
• Plot CDF

Example:

Python3

import numpy as np

import matplotlib.pyplot as plt

import pandas as pd

%matplotlib inline

N = 500

data = np.random.randn(N)

count, bins_count = np.histogram(data, bins=10)

pdf = count / sum(count)

cdf = np.cumsum(pdf)

plt.plot(bins_count[1:], pdf, color="red", label="PDF")

plt.plot(bins_count[1:], cdf, label="CDF")

plt.legend()

Output:

Histogram plot of the PDF and CDF :

Plotted CDF:

CDF plotting

Method 2: Data sort 

This method depicts how CDF can be calculated and plotted using sorted data. For this, we first sort the data and then handle further calculations.

Approach

• Import module
• Declare number of data points
• Create data
• Sort data in ascending order
• Get CDF
• Plot CDF
• Display plot

Example:

Python3

import numpy as np

import matplotlib.pyplot as plt

import pandas as pd

%matplotlib inline

N = 500

data = np.random.randn(N)

x = np.sort(data)

y = np.arange(N) / float(N)

plt.xlabel('x-axis')

plt.ylabel('y-axis')

plt.title('CDF using sorting the data')

plt.plot(x, y, marker='o')

Output: