1. What is 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 probability density function:

Where:

• \( \lambda \) — Rate parameter (events per unit time)
• \( x \) — Value at which to evaluate the function
• \( e \) — Base of natural logarithm (~2.71828)

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

3. Importance of Probability Density

Details: The probability density function helps understand the likelihood of different waiting times between events in exponential processes, crucial for reliability analysis and system modeling.

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

Q2: How does λ affect the distribution?
A: Higher λ means events occur more frequently, making shorter intervals more likely. The mean is 1/λ.

Q3: What's the relationship with Poisson distribution?
A: If events follow a Poisson process with rate λ, the waiting times between events follow an exponential distribution with parameter λ.

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: How to calculate cumulative probability?
A: The cumulative distribution function is \( F(x) = 1 - e^{-\lambda x} \), giving the probability of an event occurring before time x.