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 is widely used in reliability engineering and survival analysis.

2. How Does the Calculator Work?

The calculator uses the exponential distribution formula:

Where:

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

Explanation: The formula gives the probability density at point x for a process with rate parameter λ.

3. Importance of Probability Density Calculation

Details: The exponential distribution is fundamental for modeling waiting times, failure rates, and other time-to-event data in various fields including engineering, physics, and biology.

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: λ represents the average number of events per unit time. Higher λ means events occur more frequently.

Q2: What are typical applications of exponential distribution?
A: Modeling radioactive decay, equipment failure times, service times in queuing systems, and time between phone calls.

Q3: How is this related to Poisson distribution?
A: If events follow a Poisson process, the time between events follows an exponential distribution.

Q4: What is the mean of an exponential distribution?
A: The mean is 1/λ, which is the average waiting time between events.

Q5: 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.