Exponential Distribution Calculator Iowa

Exponential Distribution PDF:

\[ f(x) = \lambda e^{-\lambda x} \]

Probability Density (PDF):

Cumulative Probability (CDF):

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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 commonly used to model waiting times, lifetimes, and reliability analysis.

2. How Does the Calculator Work?

The calculator uses the exponential probability density function (PDF) and cumulative distribution function (CDF):

\[ PDF: f(x) = \lambda e^{-\lambda x} \] \[ CDF: F(x) = 1 - e^{-\lambda x} \]

Where:

\( \lambda \) — Rate parameter (events per unit time)
\( x \) — Time variable
\( e \) — Base of natural logarithm (~2.71828)

Explanation: The PDF gives the probability density at point x, while the CDF gives the probability that the random variable takes a value less than or equal to x.

3. Importance of Exponential Distribution

Details: The exponential distribution is fundamental in reliability engineering, queuing theory, and survival analysis. It has the memoryless property, meaning the probability of an event occurring in the next 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 time value x (must be non-negative). The calculator will output both the probability density (PDF) and cumulative probability (CDF).

5. Frequently Asked Questions (FAQ)

Q1: What are typical applications of exponential distribution?
A: Modeling time between phone calls, radioactive decay, machine component lifetimes, and time between customer arrivals.

Q2: How is the mean related to the rate parameter?
A: The mean (expected value) is 1/λ. For example, if λ=0.5 events/minute, the mean time between events is 2 minutes.

Q3: What is the memoryless property?
A: P(X > s + t | X > s) = P(X > t) for all s, t > 0. The time already waited doesn't affect future waiting time.

Q4: How does this differ from Poisson distribution?
A: Poisson counts events in fixed intervals, while exponential measures time between events. They're related but model different aspects.

Q5: What are the limitations of exponential distribution?
A: It assumes constant hazard rate, which may not match real-world scenarios where failure rates change over time.