Counting the uncountable

How many emails will you receive in the next hour? How many typos on a page? How many cars pass a checkpoint in 10 minutes?

What do you think?

A call center gets an average of 5 calls per minute. What's the probability of getting exactly 0 calls in a given minute?

From Binomial to Poisson

The Poisson distribution arises naturally from the Binomial. Imagine dividing time into tiny slices:

In 1 minute, the probability of a call is high
In each second (60 slices), the probability per slice is small
In each millisecond (60,000 slices), the probability per slice is tiny

As we slice finer and finer, keeping $n \cdot p = \lambda$ constant, the Binomial $(n, p)$ converges to the Poisson $(\lambda)$ .

Binomial → Poisson

Rate (λ): 3.0

115

Slicing granularity

Binom(10, 0.3000)

Pois(3)

0.300000

Mean

3.000

Variance

2.100

TV distance from Poisson: 8.64%

The Poisson is the limit case: infinitely many trials, each with an infinitesimally small probability, summing to a finite rate $\lambda$ .

The Poisson distribution

Poisson Distribution

A random variable $X$ follows a Poisson distribution with rate $\lambda > 0$ if: $P(X = k) = \Large\frac{e^{-\lambda} \lambda^k}{k!}, \quad \normalsize k = 0, 1, 2, \ldots$ We write $X \sim \text{Pois}(\lambda)$ .

The PMF has a clear structure:

$e^{-\lambda}$ ensures the probabilities sum to 1
$\lambda^k$ grows with $k$ (more events becomes more likely up to a point)
$k!$ in the denominator eventually dominates (very high counts become rare)

Explore how the Poisson PMF shape changes with λ — notice how mean always equals variance:

Poisson Distribution Explorer

Mode

Rate parameter λ: 4.0

0.515

P(X = k) = e^−λ · λ^k / k!

k (number of events)

E[X] = λ

4.0

Var(X) = λ

4.0

SD(X)

2.00

Mode

The Poisson signature: mean = variance = λ

Key properties

Poisson Mean and Variance

If $X \sim \text{Pois}(\lambda)$ : $E[X] = \lambda, \qquad \text{Var}(X) = \lambda$

Mean equals variance — this is the signature of the Poisson. If your data has mean ≈ variance, a Poisson model may be appropriate. If variance ≫ mean, look elsewhere.

Rate $\lambda$	Use case
0.5	Earthquakes per year in a city
2	Typos per page
5	Calls per minute
100	Website hits per second

Computing Poisson probabilities

If λ = 3, what is P(X = 0)? (to 3 decimal places) (decimal to 3 places, e.g. 0.456)

If λ = 3, what is P(X = 3)? (to 3 decimal places) (decimal to 3 places, e.g. 0.456)

When to use the Poisson

The Poisson is the right model when:

Events occur independently: One email arriving doesn't affect the next
Constant rate: The average rate doesn't change over time
No simultaneous events: In a small enough time window, at most one event occurs
Counting events in an interval: Time, space, or any "exposure" measure

Generate random events on a timeline at rate λ:

Poisson Timeline

Rate (λ): 3

120

00 events1

Observations

Avg count

0.00

Classic examples:

Number of accidents at an intersection per month
Number of mutations in a stretch of DNA
Number of photons hitting a detector per second
Number of customers entering a store per hour

What do you think?

A forest has an average of 2 lightning strikes per week. What's the probability of 0 strikes in a given week?

Enter a decimal to 3 places, e.g. 0.456

The Poisson approximation to the Binomial

Poisson Approximation

When $n$ is large, $p$ is small, and $\lambda = np$ is moderate: $\text{Binom}(n, p) \approx \text{Pois}(\lambda = np)$

This is incredibly useful for computation. Instead of calculating $\binom{10000}{3} \cdot (0.0003)^3 \cdot (0.9997)^{9997}$ , just use $\text{Pois}(3)$ .

Rule of thumb: The approximation works well when $n \geq 100$ and $p \leq 0.01$ .

1000 people, each with P=0.002 of a rare disease. Approx. P(exactly 2 cases)? (decimal to 3 places, e.g. 0.456)

See the convergence in action — fix λ and increase n to watch the Binomial PMF morph into the Poisson:

Binomial → Poisson Convergence

Fix λ = np and increase n to watch Bin(n, λ/n) converge to Pois(λ)

λ = 3: 3.0

110

n = 10 (p = λ/n = 0.3000): 10

3200

Bin(n=10, p=0.3000) Pois(λ=3)

p = λ/n

0.300000

TV distance

0.0864

The distributions differ noticeably. Slide n rightward to watch the convergence happen.

Poisson sums

The sum of independent Poissons is Poisson.

Sum of Poissons

If $X \sim \text{Pois}(\lambda_1)$ and $Y \sim \text{Pois}(\lambda_2)$ are independent, then: $X + Y \sim \text{Pois}(\lambda_1 + \lambda_2)$

If Store A gets 3 customers/hour and Store B gets 5 customers/hour (independently), the combined total is Poisson with rate 8/hour.

Poisson vs. other distributions

Distribution	Question	Parameters
Binomial	How many successes in $n$ trials?	$n$ , $p$
Geometric	How many trials until first success?	$p$
Poisson	How many events in an interval?	$\lambda$

The Poisson stands out because it models counts in continuous time or space, not a fixed number of discrete trials.

Summary

Property	Formula
PMF	$P(X = k) = e^{-\lambda} \lambda^k / k!$
Mean	$E[X] = \lambda$
Variance	$\text{Var}(X) = \lambda$
Key feature	Mean = Variance
Limit of	$\text{Binom}(n, p)$ as $n \to \infty$ , $p \to 0$ , $np = \lambda$
Sum property	$\text{Pois}(\lambda_1) + \text{Pois}(\lambda_2) = \text{Pois}(\lambda_1 + \lambda_2)$

The Poisson is for rare events at a constant rate. Think: "how many times does something happen in a fixed window of time (or space)?"

Test your understanding

A website averages 10 visits/minute. What's P(exactly 10 visits in a minute)? (to 3 decimal places) (decimal to 3 places, e.g. 0.456)

Same website. What's the variance of visits per minute? (whole number)

Two independent Poisson streams with rates 4 and 7. Combined rate? (whole number)

What's next

We've now covered the main discrete distributions. Next, we cross into the continuous world — where outcomes are real numbers and probabilities become areas under curves.