Poisson & Exponential Distributions

Rare Events and Waiting Times

Poisson Distribution

$X \sim \text{Poisson}(\lambda)$ counts the number of rare events in a fixed interval, where $\lambda$ is the average rate:

$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

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

Examples: customers arriving per hour, mutations per genome, photons hitting a detector.

Exponential Distribution

$T \sim \text{Exponential}(\lambda)$ models the waiting time between consecutive Poisson events:

$f(t) = \lambda e^{-\lambda t}, \quad t \geq 0$

$F(t) = P(T \leq t) = 1 - e^{-\lambda t}$

$E[T] = \frac{1}{\lambda}, \qquad \text{Var}(T) = \frac{1}{\lambda^2}$

Memoryless property: $P(T > s + t \mid T > s) = P(T > t)$. The distribution forgets how long you have already waited.

import math

lam = 2.0  # 2 events per hour on average

# P(exactly 2 events in one hour)
pmf = math.exp(-lam) * lam**2 / math.factorial(2)
print(round(pmf, 4))   # 0.2707

# P(wait ≤ 1 hour for next event)
cdf = 1 - math.exp(-lam * 1.0)
print(round(cdf, 4))   # 0.8647

Your Task

Implement poisson_and_exponential(lam, k, x) that prints:

1. Poisson PMF: $P(K=k)$
2. Exponential CDF: $P(T \leq x)$
3. Exponential mean: $1/\lambda$

All rounded to 4 decimal places.