Modeling Waiting Times

Probability · Axiom Academy

Discover the exponential distribution and its surprising memoryless property. ⏱ Step 1: Random Customer Arrivals You run a coffee shop. Customers arrive randomly, but on average, you get 1 customer every 2 minutes. Click the button to simulate arrivals and see how long you wait between customers. Adjust the rate parameter (λ) to see how the exponential distribution changes. Higher λ means events happen more frequently (shorter average wait times). Step 3: The Memoryless Property Here's something shocking: if you've already waited 5 minutes without a customer arriving, the probability of waiting another 5 minutes is the same as if you just started! The exponential distribution "forgets" how long you've already waited. Step 4: Where Does This Appear? The exponential distribution models waiting times in countless real-world scenarios. Click each application to see an example. Models the waiting time until the next event in a Poisson process. Characterized by a single parameter λ (rate), with PDF f(x) = λe^(-λx) for x ≥ 0. Mean = 1/λ, Variance = 1/λ². The exponential distribution is the only continuous distribution with the memoryless property: P(X > s + t | X > s) = P(X > t). This makes it perfect for modeling truly random, independent events. If the number of events in a fixed time follows a Poisson(λt) distribution, then the waiting time between events follows Exponential(λ). These two distributions are intimately connected!

This is the written version of the interactive lesson above. See the full Probability course.