📘 Session 4.7 – Exponential Distribution

📖 What Is the Exponential Distribution?

The exponential distribution describes the amount of time we wait between random events that occur at a constant average rate. It is most commonly used to model the time until the next event in a Poisson process.

💡 When to Use It

• When you are measuring time, distance, or space between events.
• The events happen randomly and independently.
• The events occur at a constant average rate.

📌 Common Examples

• ⏱️ Time between Wi-Fi dropouts in a student lounge
• 💡 Time until a light bulb burns out
• 🧍 Time between customers arriving at a help desk
• 🔬 Time between lab machine failures
• 🚌 Time until the next bus arrives (if arrivals are random)

🧠 Key Properties

• PDF: f(x) = λ e−λx for x ≥ 0
• CDF: F(x) = 1 − e−λx
• Mean (μ): 1 / λ    Standard Deviation (σ): 1 / λ

🧪 Interpretation Tip

If something happens on average 5 times per hour (λ = 5/hour), then the average time between events is 1/5 hour = 12 minutes.

🔁 Memoryless Property

The exponential distribution is unique because it has no memory. The chance of the next event occurring in the next x minutes is the same—no matter how long you’ve already waited.

Formula: P(X < t₁ + t₂ | X > t₁) = P(X < t₂)

🧮 What Is λ?

• λ (lambda) is the rate: how often something happens per unit time
• If λ = 10 per hour → ≈1 event every 6 minutes
• If λ = 2 per minute → ≈1 event every 30 seconds

📦 In Summary

• Events happen randomly but at a steady average rate
• You measure time or distance until the next event
• The waiting-time curve decays: many short waits, fewer long waits

📌 Definition

The exponential distribution models the time between events in a Poisson process. Its probability density function (PDF) is:

f(x) = λ e^−λx, for x ≥ 0

Mean: E(X) = 1/λ    Variance: Var(X) = 1/λ²

📈 Graph of Exponential Distributions

🧮 Probability Calculator (with shaded PDF)

🔍 Real Example – Computer Log-ons

Problem: Log-ons occur at a rate of 25 per hour. What is the probability of waiting more than 6 minutes (0.1 hour) for the next log-on?

🧠 Step-by-Step Explanation

• Step 1: We want P(X > 0.1).
• Step 2: λ = 25 per hour → mean wait = 1/25 = 0.04 hours.
• Step 3: Use complement of the CDF: P(X > x) = 1 − P(X ≤ x).
• Step 4: For exponential, F(x) = 1 − e−λx, so
  1. P(X > 0.1) = 1 − P(X ≤ 0.1)
  2. = 1 − F(0.1)
  3. = 1 − (1 − e−λ·0.1)
  4. = e−λ·0.1 = e−25·0.1 = e−2.5 ≈ 0.08208
• Answer: ≈ 8.2%.

✅ Key Concept Reminder

This uses the complement of the CDF (survival): P(X > x) = e−λx. The PDF is the curve’s height, not a probability by itself.

📘 Lack of Memory Property – Explained

Property: P(X < t₁+t₂ | X > t₁) = P(X < t₂)

🧪 Real-World Example (Geiger Counter)

• Mean time between detections E(X) = 1.4 minutes → λ = 1/1.4 ≈ 0.714 per minute.
• Probability of a detection within 0.5 minutes: 1 − e−0.714×0.5 ≈ 0.30.
• Even after waiting 3 minutes already, the chance in the next 0.5 minutes is still ≈ 30%.

This is why the exponential is called “memoryless.”

🎓 Practice – Wi-Fi Dropout Times

Scenario: On average, the Wi-Fi drops once every 20 minutes. What’s the probability it stays up for a 10-minute video?

Show Answer (Step-by-Step)

📥 Excel Version (with full steps)

⬇️ Download: Exponential_Probability_Calculator.xlsx

Excel-style Formulas (live preview from inputs)

Cell-based versions (match the downloadable sheet)

• λ in B5, x in B6, a in B7, b in B8
• P(X > x): =EXP(-B5*B6) and (complement) =1-(1-EXP(-B5*B6))
• P(X ≤ x): =1-EXP(-B5*B6)
• P(a < X < b): =EXP(-B5*B7)-EXP(-B5*B8)
• Mean: =1/B5 ; Variance: =1/B5^2
• Poisson link (k=0 in time x): =EXP(-B5*B6)

🔗 How the Exponential Connects to the Poisson Process (FYI)

In a Poisson process with rate λ, the number of events in a window of length t is Poisson with mean λt. The probability of zero events in that window equals the probability that the waiting time to the first event exceeds t: P(N(t)=0) = e^{−λt} = P(T > t)

See a quick simulated timeline of arrivals

# arrivals to show Resimulate