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, reliability analysis, and radioactive decay. It's memoryless, meaning the probability of an event occurring in the next interval is independent of how much time has already elapsed.

4. Using the Calculator

Tips: Enter the rate parameter λ (must be positive) and the time value x (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 is 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 is the memoryless property?
A: The probability of waiting at least t more time is the same no matter how long you've already waited.

Q4: What are typical applications?
A: Modeling time between phone calls, radioactive decay, service times in queuing systems, and equipment failure times.

Q5: How is the mean related to λ?
A: The mean of an exponential distribution is 1/λ, and the standard deviation is also 1/λ.