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 often used to model waiting times, lifetimes, and other time-to-event data.

2. How Does the Calculator Work?

The calculator uses the exponential probability density function:

Where:

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

Explanation: The function gives the relative likelihood of a random variable taking on a given value in an exponential distribution.

3. Importance of Probability Density Calculation

Details: Calculating probability density is essential for understanding the behavior of exponential processes, which appear in reliability analysis, queuing theory, and survival analysis.

4. Using the Calculator

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

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, call center wait times, equipment failure times, and inter-arrival times in queues.

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

Q5: Can this calculator handle vector inputs?
A: This version calculates for single x values. For multiple values, you would need to calculate each point separately.