The time spent waiting between events is often

The time spent waiting between events is often modeled using the exponential distribution. For example, suppose that an average of 30 customers per hour arrive at a store and the time between arrivals is exponentially distributed.

Answer

⏱️ Exponential Distribution – Modeling Customer Arrival Times

📘 Scenario Overview

When events (like customer arrivals) happen continuously and independently at a constant average rate, we can use the exponential distribution to model the time between events.

🔍 Given Data

Arrival rate (λ) = 30 customers per hour
We want to model the time between arrivals, which follows an exponential distribution.

📐 Step 1: Understand the Exponential PDF

The probability density function (PDF) of the exponential distribution is:
f(t) = λe^−λt, where t ≥ 0

Here, t is the time between arrivals, and λ is the average number of arrivals per time unit (hour).

🔄 Step 2: Convert λ to Time Units

Since λ = 30 customers/hour, we can convert it to minutes for convenience:

λ = 30 per hour = 30 ÷ 60 = 0.5 per minute

So, on average, one customer arrives every 2 minutes.

🧮 Step 3: Calculating Probabilities

To find the probability that the time between customer arrivals is less than or greater than a certain time t, use the cumulative distribution function (CDF):

P(T ≤ t) = 1 − e^−λt
P(T > t) = e^−λt

Example: What is the probability that the next customer arrives in less than 3 minutes?

λ = 0.5, t = 3
P(T ≤ 3) = 1 − e^−0.5×3 = 1 − e^−1.5 ≈ 1 − 0.2231 = 0.7769
➤ There is approximately a 77.69% chance a customer arrives within 3 minutes.

✅ Summary

Distribution type: Exponential
Mean time between arrivals: 2 minutes
λ (per minute): 0.5
Useful for: Modeling wait times, system reliability, service lines, etc.

💡 Tip: The exponential distribution is memoryless, meaning the probability of an event occurring in the next t minutes is independent of how long you’ve already waited.

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