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From Discrete to Continuous Waiting Times
The Poisson-Exponential Connection
In Chapter 2, you learned the Poisson distribution counts events in a time interval. Now we ask a different question:
How long do we wait between events?
Two Sides of the Same Coin
Poisson (Counting)
A store gets 3 customers/hour
X = # customers in 1 hour
X ~ Poisson(λ = 3)
Exponential (Waiting)
How long until the next customer?
T = time between arrivals
T ~ Exp(θ = 1/3 hour)
Key Insight: The exponential distribution is the continuous analog of the geometric distribution — both model "waiting until the first success." Geometric counts trials, exponential measures time.