# Poisson and Exponential Distributions

> The Poisson distribution models how many rare events occur in a period, like large moves or trade arrivals. Learn the formula, examples and its limits.

Source: https://learn.tradelabsai.com/math/poisson-distribution/  
Track: Math and Statistics · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Poisson and Exponential Distributions", https://learn.tradelabsai.com/math/poisson-distribution/

The Poisson distribution describes how many times an event happens in a fixed period when events occur independently at a constant average rate. It is used to model the number of trades arriving in a second, the number of large price jumps in a year, the number of defaults in a portfolio or the number of times a stop is hit. It is simple, with a single parameter, and it is the starting point for many models of event arrivals in markets.

## The formula

```
P(k events) = (λ^k × e^(-λ)) / k!
```

- **λ (lambda):** the average number of events in the period.
- **k:** the number of events (0, 1, 2, ...).

The mean and variance of a Poisson distribution are both λ.

## Worked examples

**Example: Large daily moves in a year**
Suppose a stock index historically has, on average, 3 days a year with a move larger than 4%. If such days arrive independently at that rate:

- P(0 such days) = e^(minus 3) ≈ 5.0%
- P(exactly 3) = (27 × e^(minus 3)) / 6 ≈ 22.4%
- P(6 or more) ≈ 8.4%

In reality, large moves cluster in turbulent periods, so years with many big moves happen more often than this simple model suggests. See [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/) and [GARCH](https://learn.tradelabsai.com/math/garch/).

**Example: Order arrivals**
A market maker sees an average of 2 market orders per second in a stock. The probability of no orders in a given second is e^(minus 2) ≈ 13.5%; the probability of 5 or more is about 5.3%. Models of order flow often start with Poisson arrivals and then add clustering. See [Market Making](https://learn.tradelabsai.com/strategies/market-making/).

## Relationship to other distributions

| Distribution | Relationship |
|---|---|
| Binomial | Poisson approximates the binomial when the number of trials is large and the probability small (λ = n × p). See [Binomial and Bernoulli Distributions](https://learn.tradelabsai.com/math/binomial-distribution/) |
| Exponential | The time between Poisson events follows an exponential distribution with mean 1 / λ |
| Normal | For large λ, the Poisson looks approximately normal |

## Uses in finance

| Use | Example |
|---|---|
| Jump models | Merton's jump diffusion model (1976) adds Poisson distributed price jumps to option pricing |
| Credit risk | Counting defaults in a portfolio, with intensity models |
| Market microstructure | Arrival of orders and trades. See [The Order Book and Market Depth](https://learn.tradelabsai.com/market-structure/the-order-book-and-market-depth/) |
| Operational risk | Frequency of losses from errors or system failures. See [Operational and Model Risk](https://learn.tradelabsai.com/portfolio/operational-and-model-risk/) |
| Insurance and catastrophe bonds | Frequency of disasters |

## Overdispersion and clustering

Real market events usually show overdispersion: the variance of counts is larger than the mean, because events cluster. Large moves come in bursts during crises; orders arrive in waves after news. Models that handle this include:

- **Negative binomial distribution:** allows variance larger than the mean.
- **Hawkes processes:** self exciting processes where each event raises the chance of more events soon after, widely used for order flow and jump clustering.

## Limits

- **Constant rate assumption:** event rates change with market conditions.
- **Independence assumption:** events often trigger more events.
- **Rare event estimates:** with few historical events, λ itself is uncertain.

## Estimating the rate

The simplest estimate of λ is the number of events divided by the length of the observation period. With few events, that estimate is uncertain: if you saw 3 large moves in 2 years, the true yearly rate could plausibly be anywhere from under 1 to about 4. Using longer histories, or pooling similar assets, gives more reliable rates.

## Frequently asked questions

### What is the Poisson distribution?

A distribution describing the number of times an event occurs in a fixed period, assuming events happen independently at a constant average rate.

### How is the Poisson distribution used in trading?

To model counts of rare events such as large price jumps, defaults and order arrivals, and as a building block in jump diffusion and microstructure models.

### What are the limits of the Poisson model in markets?

Market events cluster and their rates change over time, so counts are often more variable than a Poisson model predicts.

Next, learn why extreme moves happen so often in [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/).

## Continue learning

- Next lesson: [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/)
- Previous lesson: [Binomial and Bernoulli Distributions](https://learn.tradelabsai.com/math/binomial-distribution/)
- Related: [Binomial and Bernoulli Distributions](https://learn.tradelabsai.com/math/binomial-distribution/): The binomial distribution gives the probability of a number of wins in a set of trades. Learn the formula, trading examples and its link to option trees.
- Related: [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/): Fat tails mean extreme market moves happen far more often than the normal curve predicts. Learn the evidence, the causes, how to measure them and how to manage them.
- Related: [Probability Distributions Explained](https://learn.tradelabsai.com/math/probability-distributions/): Probability distributions describe the range and likelihood of outcomes. Learn the main ones used in trading, their shapes and when each applies.
- Related: [Market Data Replay](https://learn.tradelabsai.com/programming/market-data-replay/): Market data replay feeds recorded live data back through a trading system to test, debug and benchmark it. Learn how to record, replay and stay deterministic.
- Related: [Risk of Ruin](https://learn.tradelabsai.com/risk/risk-of-ruin/): Risk of ruin is the chance that losses drain your account beyond recovery. Learn what drives it, see simulated numbers and how to keep it low.