3.3.2 Memoryless Property of Exponential Distribution

3.3.2 Memoryless Property of Exponential Distribution

Overview

This section explores the unique memoryless property of the exponential distribution, a characteristic that sets it apart from most other probability distributions.

Key Concepts

• Memoryless Property: The property that the remaining time until an event occurs is independent of any time that has already passed.
• Mathematical Formulation:
  • If X is an exponential random variable with rate λ, then for any s, t ≥ 0,
    • P(X > s + t | X > s) = P(X > t)

Detailed Explanation

Definition and Explanation

• Exponential Random Variable: Measures time until an event with a constant average rate λ, expressed as E(X) = 1/λ.
• Memoryless Example: Once a certain amount of time s has elapsed, the distribution of the remaining time until the event occurs is the same as the original distribution.

Mathematical Derivation

• Conditional Probability: Shows how the memoryless property mathematically simplifies conditional probability calculations.
• Derivation: For an exponential random variable X and s, t ≥ 0, the property is shown by:
  • P(X > s + t | X > s) = P(X > s + t) / P(X > s) = e^(-λ(s + t)) / e^(-λs) = e^(-λt) = P(X > t)

Practical Example

Scenario: Shanti's bus waiting times modeled as exponential with mean 20 minutes (λ = 0.05).

Probability Calculations:

• Less than 15 minutes: 1 - e^(-0.05 * 15) ≈ 0.528
• More than 30 minutes: e^(-0.05 * 30) ≈ 0.223
• Between 15 and 30 minutes: e^(-0.05 * 15) - e^(-0.05 * 30) ≈ 0.249

Memoryless Demonstrations:

• After waiting 10 minutes, probability of waiting another 10 minutes: e^(-0.05 * 10) ≈ 0.607
• After waiting 20 minutes, probability of waiting another 10 minutes remains 0.607, demonstrating the memoryless property.