Question

Prove that an exponential random variable has skewness two.

   Prove that an exponential random variable has skewness two.

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Arnold O. Allen 2nd Edition

Chapter 7, Problem 12

Instant Answer

Step 1

An exponential random variable \( X \) with rate \( \lambda \) has the probability density function (PDF) given by: \[ f(x) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0 \] Show more…

Show all steps

lock

Thanks for your feedback!

Prove that an exponential random variable has skewness two.

Play audio

Feedback

Powered by NumerAI

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution that is often used to model the time between independent events that occur at a constant average rate. It is defined by a single parameter and exhibits the memoryless property, meaning that the probability of an event occurring in the future is independent of the past.

Skewness

Skewness is a statistical measure that quantifies the degree of asymmetry of a probability distribution around its mean. It is calculated as the third standardized moment, and a skewness of two indicates a pronounced right skew (longer tail to the right) relative to a symmetric distribution.

Moments and Central Moments

Moments are numerical values that capture various aspects of the shape of a probability distribution, such as location, spread, and asymmetry. Central moments, which measure deviations from the mean, are particularly important; the third central moment, when standardized, provides the skewness of the distribution.

Recommended Videos

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later