rkucar / dsc-cumulative-distribution-function-lab-online-ds-sp-000

The Cumulative Distribution Function - Lab

Introduction

In the previous lesson, you learned how you can create a cumulative distribution function for discrete and continuous random variables. In this lab, you'll try to calculate a CDF for a dice roll yourself, and visualize it.

Objectives

You will be able to:

• Calculate CDF in Python for a given discrete variable with a limited set of possible values
• Visualize and inspect a CDF in order to make assumptions about the underlying data

Calculating CDF in Python

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

$$\Large F(x)= P(X \leq x)$$

So given a list of all possible values of x, We can easily calculate the CDF for a given possible value $X$ by performing the 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 dice roll), and $X$ is the value for which we want to calculate the cumulative distribution function
• Initialize a variable called count
• For all values in lst, if a value is less than or equal to $X$, add one to count - do nothing otherwise. (this will tell us the total number of values less than $X$)
• Calculate the cumulative probability of $X$ dividing count by the total number of possible values
• Round by 3 decimals and return the cumulative probability of $X$

def calculate_cdf(lst, X):
    
    pass

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

calculate_cdf(test_lst, test_X)

# 0.667

0.667

Now, use this function to calculate a CDF for each value in a dice roll so you can plot it later on.

Perform the following steps in the cell below:

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

dice_lst = None
dice_cum = None

dice_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: one list which contains the potential outcome values of your discrete distribution, and another list which contains cumulative probabilities.

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 a bar plot.

• Use dice_lst for x-axis and dice_cum for y-axis

# Your code here

Level Up (optional)

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 the following steps

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.scatter() method with np.cumsum() to calculate and plot cumulative probabilities (just like we did above).

# Your code here

Summary

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.