1. What is the Exponential Distribution?

The Exponential Distribution is a continuous probability distribution that describes the time between events in a Poisson process. It is widely used in reliability engineering, queuing theory, and survival analysis.

2. How Does the Calculator Work?

The calculator uses the Exponential PDF formula:

Where:

• \( \lambda \) — Rate parameter (events per unit time)
• \( x \) — Value at which to evaluate the PDF
• \( e \) — Euler's number (approximately 2.71828)

Explanation: The exponential distribution models the probability distribution of the time between events in a memoryless process.

3. Importance of Exponential Distribution

Details: The exponential distribution is fundamental in modeling waiting times, failure rates, and other time-to-event data where events occur continuously and independently at a constant average rate.

4. Using the Calculator

Tips: Enter the rate parameter λ (must be positive) and the value x (must be non-negative). The calculator will compute the probability density at point x.

5. Frequently Asked Questions (FAQ)

Q1: What does the rate parameter λ represent?
A: The rate parameter λ represents the average number of events per unit time. A higher λ means events occur more frequently.

Q2: What is the memoryless property?
A: The exponential distribution is memoryless, meaning the probability of an event occurring in the next time interval is independent of how much time has already elapsed.

Q3: What are typical applications of exponential distribution?
A: Modeling waiting times, radioactive decay, service times in queuing systems, and time between equipment failures.

Q4: How is exponential distribution related to Poisson distribution?
A: If events follow a Poisson process with rate λ, then the time between events follows an exponential distribution with the same rate parameter λ.

Q5: What is the relationship between mean and rate parameter?
A: The mean of an exponential distribution is 1/λ, and the standard deviation is also 1/λ.