1. What is the Exponential Distribution?

The exponential distribution describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It's widely used in reliability engineering and queuing theory.

2. How Does the Calculator Work?

The calculator uses the exponential distribution formula:

Where:

• \( \lambda \) — Rate parameter (events per unit time)
• \( x \) — Time value
• \( e \) — Euler's number (~2.71828)

Explanation: The formula calculates the probability that the time between events is less than or equal to x.

3. Importance of Exponential Distribution

Details: The exponential distribution is crucial for modeling waiting times, radioactive decay, service times in queuing systems, and reliability analysis of products.

4. Using the Calculator

Tips: Enter the rate parameter λ (must be positive) and the x value (must be non-negative). The calculator will compute the cumulative probability P(X ≤ x).

5. Frequently Asked Questions (FAQ)

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

Q2: What's the relationship between exponential and Poisson distributions?
A: If events follow a Poisson process, the time between events follows an exponential distribution.

Q3: What's the mean of an exponential distribution?
A: The mean is 1/λ, which represents the average time between events.

Q4: What's the memoryless property?
A: The exponential distribution is memoryless, meaning P(X > s + t | X > s) = P(X > t).

Q5: When is exponential distribution not appropriate?
A: When event rates vary over time or events aren't independent, other distributions may be more suitable.