• Models the number of events occurring in a fixed interval of time or space
• Occurrences happen independently from one another
• Events can’t occur and reside at the same moment it occurs
• Events occur at a constant rate over the time it’s being observed
• Examples: customer arrivals per hour, typos per page, API requests per second

Better known as the Probability Mass Function

For a Poisson random variable \(X\) with mean rate parameter \(\lambda > 0\):

\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots\]

• \(\lambda\) is the average amount of events expected to be observed per time interval
• In practice, it’s calculated as total events observed, divided by number of intervals observed

\[E[X] = \lambda \qquad \text{Var}(X) = \lambda\]

• A defining property of the Poisson distribution: the mean equals the variance
• Both are controlled by the same single number, \(\lambda\)

\[E[X] = \sum_{k=0}^{\infty} k \cdot \frac{\lambda^k e^{-\lambda}}{k!} = \lambda \sum_{k=1}^{\infty} \frac{\lambda^{k-1} e^{-\lambda}}{(k-1)!} = \lambda\]

\[\text{Var}(X) = E[X^2] - (E[X])^2 = (\lambda^2 + \lambda) - \lambda^2 = \lambda\]

• Suppose a lemonade stand receives an average of \(\lambda = 4\) customers every 30 minutes
• That’s a rate of \(\lambda = 4/3\) customers every 10 minutes
• Question: what is the probability of exactly 3 customers arriving in the next 10 minutes?

dpois(3, lambda = 4/3)

## [1] 0.1041371

set.seed(20) # only for locking RNG
sim_data <- rpois(n = 1000, lambda = 4/3)
head(sim_data, 20)

##  [1] 3 2 1 1 4 4 0 0 1 1 2 2 0 2 0 1 1 0 1 2

• rpois() generates random draws from a Poisson distribution
• Here, 1000 simulated “10-minute windows” of customer counts

• Bar height at each \(k\) shows the probability of exactly \(k\) customers arriving
• Built using \(\lambda = 4/3\), matching the 10-minute lemonade stand rate
• Notice the peak sits around \(k = 1\), since \(\lambda \approx 1.33\) is the average

• The Poisson distribution models counts independent events in a fixed interval
• Fully described by a single parameter \(\lambda\), which equals both the mean and the variance
• Used across many fields: biology, network traffic, sport statistics