Poisson distribution

The Poisson distribution counts how many times a rare event occurs in a fixed interval — radioactive decays per second, calls arriving at a switchboard per minute, typos per page. One parameter, $λ$ , does all the work: it is simultaneously the mean and the variance.

P (k; λ) = \frac{λ ^{k} e ^{- λ}}{k !}

λ (rate) 4.0 n (samples) 500

Dark bars: PMF

P (k; λ)

. Light bars: empirical frequencies from 500 samples. Dashed: Normal(

λ, λ

) approximation.

What to notice

Small $λ$ produces a strongly right-skewed shape — rare events cluster near zero with a long tail.
Growing $λ$ makes the bars more symmetric and bell-shaped. The dashed Normal( $λ$ , $λ$ ) curve hugs the PMF tightly above λ ≈ 10 — that’s the Central Limit Theorem at work on independent counts.
Adding samples smooths the light bars toward the dark PMF bars, illustrating the Law of Large Numbers.

Connection to the exponential

If events arrive at rate $λ$ per unit time as a Poisson process, the waiting time between consecutive arrivals follows an Exponential( $λ$ ) distribution. The two distributions are two views of the same underlying process.