In the previous lesson we saw how we can use a discrete random variable used for modeling fair die having a uniform probabilities for all possible values. In this lab, we shall try to calculate a cdf for this variable and visualize it for inspection.

You will be able to:

• Calculate the cdf for a given discrete random variable
• Visualize the cdf using matplotlib

Recall the formula for calculate the cumulative probability from previous lesson:

So given a list of all possible values of x, We can easily calculate the cdf for a given possible value (X) by performing following steps:

• Build a function calculate_cdf(lst,X), where lst is a list of all possible values in a discrete variable x (6 values for a die roll), and X is the value for which we want to calculate the cumulative distribution function.
• Initialize a count variable
• for all values in lst, if a value is less than X, add one to count - do nothing otherwise. (this will tell us total number of values less than X)
• Calculate the cumulative probability of X dividing the count with total possible values
• Round off by 3 decimals and return the cumulative probability of X.

def calculate_cdf(lst, X):
    count = 0
    for i in lst:
        if i <= X:
            count += 1
        #print("X is: {}, i is {}, count-cdf: {}".format(X, i, count))
    cdf = count / len(lst)
    return round(cdf, 3)

# test data
test_lst = [1,2,3]
test_X = 2

calculate_cdf(test_lst, test_X)

# 0.667

0.667

Let's now use above function to calculate a cdf for each value in a die roll with an intention of plotting it.

Perform following steps in the cell below:

• Create a list die_lst with all possible values of a fair die
• Initialize an empty list die_cum for storing cumulative probabilities for these values.
• For each value in the die_lst calculate its cumulative probability using the function above and store in die_cum list.

die_lst = [1,2,3,4,5,6]
die_cum = []
for i in range(len(die_lst)):
    die_cum.append(calculate_cdf(die_lst, die_lst[i]))
die_cum

# [0.167, 0.333, 0.5, 0.667, 0.833, 1.0]

[0.167, 0.333, 0.5, 0.667, 0.833, 1.0]

cdfs are implemented with two sorted lists: xs, which contains the values, and ps, which contains the cumulative probabilities for xs.

Following this, we now have a list of possible values, and a second list containing cumulative probabilities for each value. Let's go ahead and plot these values in matplotlib using the stem plot.

• Use die_lst for x-axis and die_cum for y-axis

Text(0,0.5,'Cumulative Probabilities')

import matplotlib.pyplot as plt
plt.style.use('ggplot')
plt.stem(die_lst, die_cum, '-.', 'ro', 'g-')
plt.show()

cdfs (and pmfs) can be calculated using built in numpy and matplotlib methods. So we don't have create custom functions to calculate these. We can draw a histogram styled cdf as shown below using following methods.

You would need to perform these steps

• Use np.histogram() to automatically calculate the histogram with probabilities. Here is numpy histogram documentation to help you dig deeper.

• Use plt.step() method with np.cumsum() to calculate and plot cumulative probabilities (just like we did above).

import numpy as np

die_hist = np.histogram(die_lst,bins=6, range=(1,7), density=True)
die_hist

(array([0.16666667, 0.16666667, 0.16666667, 0.16666667, 0.16666667,
        0.16666667]), array([1., 2., 3., 4., 5., 6., 7.]))

plt.step(x=die_hist[1][0:6], y=np.cumsum(die_hist[0]))
plt.title("Die Roll - Cumulative Distribuion Function")
plt.xlabel("Die Values")
plt.ylabel("Cum Sum Probability")

Text(0, 0.5, 'Cum Sum Probability')

In this lesson we looked at developing a cdf a percentile function of a discrete random variable. We looked at how to calculate and visualize a cdf. This technique can also be applied to continuous random variables which we shall see later in this section.