1. What is 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 distribution formula:

Where:

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

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

3. Importance of Exponential Distribution

Details: Exponential distribution is crucial for modeling time-to-failure in reliability analysis, inter-arrival times in queuing systems, and survival times in medical research.

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 probability density function value at the given x.

5. Frequently Asked Questions (FAQ)

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

Q2: Is exponential distribution memoryless?
A: Yes, the exponential distribution has the memoryless property, 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?
A: Modeling time between phone calls at a call center, time between failures of mechanical systems, and radioactive decay processes.

Q4: How is it 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.

Q5: What is the mean and variance?
A: Mean = 1/λ, Variance = 1/λ². The standard deviation equals the mean.