Poisson Distribution

The Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events happen independently at a constant average rate $\lambda$.

PMF

The probability of observing exactly $k$ events:

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

• Mean: $E[X] = \lambda$
• Variance: $\text{Var}(X) = \lambda$

Real-World Examples

Examples

For $\lambda = 1$: $P(X=0) = e^{-1} \approx 0.3679 \qquad P(X=1) = e^{-1} \approx 0.3679$

For $\lambda = 3$: $P(X=3) = \frac{e^{-3} \cdot 27}{6} \approx 0.2240$

Implementation

import math

def poisson_pmf(lam, k):
    return round(math.exp(-lam) * lam**k / math.factorial(k), 4)

Relationship to Binomial

When $n$ is large and $p$ is small with $np = \lambda$, the binomial distribution $B(n,p)$ approaches Poisson$( \lambda)$. This is why Poisson appears in rare-event scenarios.

Your Task

Implement poisson_pmf(lam, k) that returns $P(X=k)$ for a Poisson distribution with rate $\lambda$, rounded to 4 decimal places.