1. What is Exponential Probability Density?

The exponential probability density function describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is commonly used to model waiting times, failure times, 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 PDF
• \( e \) — Euler's number (approximately 2.71828)

Explanation: The function gives the probability density at point x for an exponential distribution with rate parameter λ.

3. Importance of Exponential Distribution

Details: The exponential distribution is fundamental in reliability engineering, queuing theory, and survival analysis. It is memoryless, meaning the probability of an event occurring in the next time 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 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: The rate parameter λ represents the average number of events per unit time. A higher λ means events occur more frequently.

Q2: What is the relationship between PDF and CDF?
A: The probability density function (PDF) gives the density at a point, while the cumulative distribution function (CDF) gives the probability that the random variable is less than or equal to a value.

Q3: What are typical applications of exponential distribution?
A: Commonly used to model waiting times, time between phone calls, lifespan of electronic components, and time between earthquakes.

Q4: What does the memoryless property mean?
A: The memoryless property means that the probability of an event occurring in the next time interval is independent of how much time has already passed.

Q5: How is exponential distribution related to Poisson distribution?
A: The exponential distribution models the time between events in a Poisson process, while the Poisson distribution models the number of events in a fixed interval.