CHAPTER 3A — DISCRETE PROBABILITY
DISTRIBUTIONS (UNIVARIATE)
This chapter has 4 big parts:
1. Discrete Random Variables
2. Probability Distributions (PMF / PDF for discrete)
3. Distribution Functions (CDF)
4. Examples:
o Dice
o Coin (Binomial)
o Socks (Hypergeometric)
o Accidents (Given CDF → find pdf)
We will go step by step.
PART 1: DISCRETE RANDOM VARIABLES
What is a Sample Space?
The sample space (S) is:
The set of all possible outcomes of an experiment.
Example: Roll two dice
Each die has numbers 1 to 6.
If you roll two dice:
Possible outcomes are ordered pairs:
(1,1), (1,2), …, (6,6)
Total outcomes:
6 × 6 = 36
So:
S = { (1,1), (1,2), … , (6,6) }
There are 36 sample points.
What is a Random Variable?
A random variable:
A function that assigns a number to each outcome in the sample space.
Very important:
• Random variables are written in CAPITAL letters (X)
• Values are written in lowercase (x)
, Dice Example
Define:
X = total number of points when rolling 2 dice
So:
If outcome is (3,5), then:
X=3+5=8
If outcome is (6,6), then:
X = 12
What is the Range of X?
Minimum total:
1+1=2
Maximum total:
6 + 6 = 12
So:
Range of X = {2, 3, 4, … , 12}
This is a finite set, so X is a discrete random variable.
What is a Discrete Random Variable?
A random variable is discrete if:
• Its values are finite
OR
• Countably infinite (like 1,2,3,4,...)
Examples:
• Number of bedrooms (finite)
• Number of coin tosses until heads (countable infinite)
DISTRIBUTIONS (UNIVARIATE)
This chapter has 4 big parts:
1. Discrete Random Variables
2. Probability Distributions (PMF / PDF for discrete)
3. Distribution Functions (CDF)
4. Examples:
o Dice
o Coin (Binomial)
o Socks (Hypergeometric)
o Accidents (Given CDF → find pdf)
We will go step by step.
PART 1: DISCRETE RANDOM VARIABLES
What is a Sample Space?
The sample space (S) is:
The set of all possible outcomes of an experiment.
Example: Roll two dice
Each die has numbers 1 to 6.
If you roll two dice:
Possible outcomes are ordered pairs:
(1,1), (1,2), …, (6,6)
Total outcomes:
6 × 6 = 36
So:
S = { (1,1), (1,2), … , (6,6) }
There are 36 sample points.
What is a Random Variable?
A random variable:
A function that assigns a number to each outcome in the sample space.
Very important:
• Random variables are written in CAPITAL letters (X)
• Values are written in lowercase (x)
, Dice Example
Define:
X = total number of points when rolling 2 dice
So:
If outcome is (3,5), then:
X=3+5=8
If outcome is (6,6), then:
X = 12
What is the Range of X?
Minimum total:
1+1=2
Maximum total:
6 + 6 = 12
So:
Range of X = {2, 3, 4, … , 12}
This is a finite set, so X is a discrete random variable.
What is a Discrete Random Variable?
A random variable is discrete if:
• Its values are finite
OR
• Countably infinite (like 1,2,3,4,...)
Examples:
• Number of bedrooms (finite)
• Number of coin tosses until heads (countable infinite)
