Introduction to the Poisson Distribution
Imagine it's Week 14 of the 2023 NFL season, and you're analyzing a prop on Minnesota Vikings wide receiver Justin Jefferson's receiving touchdowns against the Chicago Bears. The line is set at Over 0.5 touchdowns (-150), Under 0.5 (+120). Jefferson has been on fire: he's averaging 0.85 touchdowns per game over his last 10 starts, and the Bears rank 28th in red zone defense. The Vikings are 7-point favorites, suggesting multiple scoring opportunities.
You pull up your notes from Chapter 7, where we used the normal distribution to model continuous stats like passing yards. But touchdowns are different. They're discrete counts: 0, 1, 2, 3—never 1.7 or 2.3. They're also relatively rare events within a game's context. The normal distribution (a smooth bell curve spanning all real numbers) doesn't match that reality.
Enter the Poisson distribution—a tool designed for modeling rare, discrete events.
What Is the Poisson Distribution?
In sports betting, the Poisson distribution helps answer questions like:
- "What's the probability of exactly 2 touchdowns?"
- "What are the odds of 8+ strikeouts?"
- "How likely is a player to score 0 goals?"
Key Insight
The Poisson distribution turns a single input (λ) into a full set of probabilities for discrete counts—exactly what you need for many prop markets.
The key parameter is lambda (λ), the expected number of occurrences in the interval you care about (for example, touchdowns per game). If you can estimate λ better than the market, you have an edge.
Why Not Use the Normal Distribution?
For continuous stats like passing yards, points, or receiving yards, the normal distribution works beautifully. But count-based props have fundamentally different characteristics:
| Normal Distribution | Poisson Distribution |
|---|---|
| Continuous outcomes (can be any decimal) | Discrete counts (0, 1, 2, 3...) |
| Symmetric bell curve | Right-skewed for small λ |
| Can go negative (theoretically) | Bounded at zero |
| Best for high-volume stats | Best for rare events |
Examples by Distribution Type
Use Normal Distribution:
- NFL Passing Yards
- NBA Points (starter, 34+ minutes)
- MLB Pitcher Outs
Use Poisson Distribution:
- Touchdowns (0, 1, 2, 3...)
- Strikeouts (0, 1, 2, 3...)
- Three-pointers made
- Goals (hockey, soccer)
- Home runs
- Assists
Tip
Use Poisson when outcomes are small whole numbers and 0/1/2 are common; use Normal when volume is high and outcomes behave more continuously.
Building on Previous Chapters
This chapter builds directly on:
- Chapter 4's expected value framework — We'll convert Poisson probabilities into EV calculations
- Chapter 7's probability calculations — Where normals excel at continuous outcomes, Poisson is built for event counts
A Word of Caution
The Poisson model assumes:
- Independence — Each scoring chance doesn't depend on the last
- Constant average rate — λ doesn't change during the game
Reality is messier. Game script shifts, defenses adjust, injuries happen. The goal here isn't perfection—it's a practical baseline you can beat by estimating λ better than the market.
Warning
If a player's role, minutes, pitch count, or game script can swing dramatically, you may still use Poisson, but you should demand a larger edge before betting.
Where Poisson Props Show Up
Count-based props show up everywhere across sports:
NFL
- Passing/Receiving/Rushing Touchdowns
- Receptions
- Sacks
NBA
- Three-pointers made
- Steals
- Blocks
MLB
- Strikeouts
- Hits
- Home runs
- RBIs
NHL/Soccer
- Goals
- Assists
- Shots on goal
These markets share the same structure: you're not betting on a smooth continuous number—you're betting on how many times something happens.
📝 Exercise
Instructions
Test your understanding of when to use Poisson vs. Normal distribution.
A quarterback averages 280 passing yards per game with a standard deviation of 65. Which distribution should you use to model his passing yards prop?
Which of the following props is BEST suited for Poisson modeling?