Poisson Distribution

📈 Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming the events occur independently and at a constant average rate.

📌 Definition

Let $X \sim \text{Poisson}(\lambda)$, where $\lambda > 0$ is the expected number of events in the interval. Then:

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

📋 Key Properties

Property	Value
Parameter	$\lambda > 0$
Support	$k \in {0, 1, 2, \dots}$
Mean	$\mathbb{E}[X] = \lambda$
Variance	$\text{Var}(X) = \lambda$
Mode	$\lfloor \lambda \rfloor$ or $\lfloor \lambda \rfloor - 1$
MGF	$M_X(t) = \exp(\lambda(e^t - 1))$

🌍 Real-World Examples

1. Call Center
   Number of phone calls received in an hour.

2. Accidents
   Number of car accidents at an intersection in a week.

Notes

1. Appropriate for modeling rare events over time/space.
2. Variance equals the mean — if data shows overdispersion, consider Negative Binomial.
3. Often used in queueing theory, network traffic, and biostatistics.
4. The Poisson process assumes independence between occurrences.

🐍 Python Example

from scipy.stats import poisson

# Mean number of events (lambda)
λ = 4

# Create Poisson distribution
rv = poisson(λ)

# PMF at specific value
print("P(X=2):", rv.pmf(2))

# Sample 10 values
print("Samples:", rv.rvs(size=10))

# Mean and variance
print("Mean:", rv.mean())
print("Variance:", rv.var())