Most random variables fall into two big categories:

1. Discrete random variables
2. 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.