Most random variables fall into two big categories:
- Discrete random variables
- Continuous random variables
Everything else is a refinement of these two.
🎯 1. Discrete Random Variables
A discrete random variable takes countable values — usually integers.
Key features
- Values can be listed or counted
- Often arise from counting something
- Probabilities are assigned to individual outcomes
Examples
- Number of heads in 10 coin flips (0, 1, 2, …, 10)
- Number of customers arriving in an hour
- Number of defective items in a batch
- Result of rolling a die (1–6)
Common discrete distributions
Bernoulli (0 or 1)
Binomial (number of successes in n trials)
Geometric (trial of first success)
Poisson (counts over time/space)
🎯 2. Continuous Random Variables
A continuous random variable takes values from an interval — uncountably many possibilities.
Key features
- Values form a continuum
- Probabilities are assigned to intervals, not points
- Probability of any exact value is 0
Examples
- Height of a person
- Time until a machine fails
- Amount of rainfall
- Temperature at noon
Common continuous distributions
Normal (bell curve)
Uniform (equal density over an interval)
Exponential (waiting times)
Gamma, Beta, etc.
⭐ Bonus: Mixed Random Variables
Some random variables combine discrete and continuous parts.
Example
A machine produces:
- 90% of items with a continuous weight distribution
- 10% defective items that weigh exactly 0 grams
This variable has:
- A point mass at 0 (discrete)
- A continuous distribution for the rest
These are less common in intro courses but appear in real‑world modeling.
🎨 Intuition
Discrete = counting marbles
Continuous = measuring water in a glass
Both are random, but one is about how many, the other is about how much.