This document provides an introduction to common probability distributions used in data analytics. It discusses both discrete and continuous distributions, including the uniform, binomial, Poisson, and exponential distributions. It provides the probability mass functions (PMFs) and probability density functions (PDFs) that define each distribution, as well as explaining how to calculate the cumulative distribution function (CDF), mean, and variance for both continuous and discrete random variables.

1. Introduction to Data Analytics
Lecture: Random Variables and Probability distributions - 2
NPTEL MOOC
By
Prof. Nandan Sudarsanam, DoMS, IIT-M and
Prof. B. Ravindran, CS&E, IIT-M

2. • Uniform
• Discrete
• The six sided dice, coin toss
• Formula for pdf: 𝑓 𝑋 = 𝑥 =
1
𝑘
for all 𝑥 that belongs to a specific set with k elements
And 𝑓 𝑋 = 𝑥 = 0 for all other values of x.
• Continuous
• Number of seconds past the minute
• Exact age of a randomly selected person between the ages of 50-60
• Formula for PDF:
• What is the CDF, mean and Variance?
CDF =
𝑥−𝑎
𝑏−𝑎
Mean =
1
2
(𝑏 + 𝑎)
Variance =
1
12
(𝑏 − 𝑎)2
Common distributions
1
for a x b
(x)= b-a
0 for x<a and x>b
f
 
 
 
 
 
 
1/(b-a)

3. Common distributions
• Binomial
• What is it + Example: Toy problem
• Example Real-world: Probability of 3 out of 10 mergers. Probability of there
being 5 defective products in a batch of 20.
• Formula for PMF:
• Formula for CDF is just the summation
• It is more useful for small n’s
• Mean: np, variance: np(1-p)
k
n
k
p
p
k
n 









)
1
(

4. • Poisson
• Discrete distribution that signifies the probability of ‘x’ occurrences of a
certain event over a certain period of time or space.
• Examples: Number of defaults per month, Number of banks per square
kilometre.
• PMF (not PDF)
• Mean and variance are 𝜆 (lambda >0)
Common distributions

 
e
k
k
!

5. Common distribution
1
(1 )k
p p


1 (1 )k
p
 

6. Common distributions
• Exponential
• The interarrival times of the Poisson distribution
• The continuous version of the geometric distribution
• Memoryless
• PDF: 𝜆𝑒−𝜆𝑥, where lambda>0
• CDF: 1-𝑒−𝜆𝑥
• Mean:
1
𝜆
• Variance:
1
𝜆2

7. Common distributions
Interarival Distribution
Count per unit interarrival
distribution
Discrete Interarrival Geometric Binomial
Continuous interarrival Exponential Poisson
Continuous Distribution
Discrete Distribution
• Parallels to the Binomial, Exponential, Geometric

8. working with distributions
• Going from PDF to CDF (continuous)
𝐹 𝑥 = 𝑓 𝑥 𝑑𝑥
𝑥
−∞
• Going from CDF to PDF (continuous)
𝑓 𝑥 =
𝑑
𝑑𝑥
𝐹(𝑥)
• Mean
• Variance/Standard deviation
1 2 ... n
x x x
x
n
  
  
1
i i
i
E x p x


    ( )
E x xf x dx


 
2
1
1
( )
1
n
i
i
s x x
N 
 

 𝜎 =
1
𝑁
(𝑥𝑖 − 𝜇)2
𝑁
𝑖=1
𝑝𝑖 = 𝜎 = 𝑥2𝑓 𝑥 𝑑𝑥 − 𝜇2
∞
−∞