Skip to content

3.3.1 Exponential Distribution

3.3.1 Exponential Distribution

Overview

This section delves into the exponential random variable, exploring its probability density function (PDF), key properties, and various real-world applications. The exponential distribution models the time between occurrences of random events with a constant average rate.

Key Concepts

  • Exponential Random Variable: Measures the time between random events, taking only positive values.

  • Probability Density Function (PDF): Defined by image

    where λ is the rate of occurrence per unit time. * Cumulative Distribution Function (CDF): Given by image

    representing the probability of an event occurring by a certain time x.

Detailed Explanation

Properties of the Exponential Distribution

  • Memorylessness: The distribution's future probabilities are not affected by any knowledge of when the last event occurred, unique to the exponential distribution.

  • Mean and Variance: Both are defined by λ, with the mean

    • E(X) = 1 / λ
    • Var(X) = 1 / λ^2

  • Skewness: The distribution is right-skewed, indicating a longer tail to the right of the mean.

Computational Formulas

  • Probability between two points (a and b): image

  • Expectation and Variance Calculations: Both integral and practical formulas highlight the simplicity and elegance of exponential distribution calculations.

Applications

  • Service Times: Modeling call center wait times or repair times.
  • Lifetime Modeling: Used for equipment and component life expectancy.
  • Queueing Theory: Useful in predicting waiting times in queues.

Practical Example

Scenario: Megha Jain's wait times at a travel service desk.

Mean Wait Time: 2 minutes, implying λ = 0.5.

Probability Queries:

  • Less than one minute: image

  • More than three minutes: image

  • Between one and three minutes:

    • P(1 < X < 3) ≈ 0.383

  • 90th percentile of waiting time: Approximately 4.605 minutes.

Ask Hive Chat Chat Icon
Hive Chat
Hi, I'm Hive Chat, an AI assistant created by CollegeHive.
How can I help you today?