1. What is the Exponential Distribution?

Definition: The exponential distribution models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.

Purpose: Used to analyze waiting times, such as the time until a machine fails or a customer arrives.

2. How Does the Calculator Work?

The calculator computes probabilities and statistics for an exponential distribution with rate parameter \(a\):

• \(P(x > X) = \exp(-aX)\): Probability that time exceeds \(X\).
• \(P(x \leq X) = 1 - \exp(-aX)\): Probability that time is at most \(X\).
• Mean (\(\mu\)) = \(1/a\): Average time between events.
• Median (\(m\)) = \(\ln(2)/a\): Time below which 50% of events occur.
• Variance (\(\sigma^2\)) = \(1/a^2\): Spread of the distribution.
• Standard Deviation (\(\sigma\)) = \(1/a\): Square root of variance.

Steps:

• Enter the rate parameter (\(a\)) and target time period (\(X\)).
• Click "Calculate" to compute probabilities and statistics.

3. When to Use the Exponential Distribution?

Use when:

• Modeling time between independent events occurring at a constant rate.
• Analyzing processes like equipment failures, customer arrivals, or radioactive decay.

4. Using the Calculator

Example: Suppose a machine fails on average once every 2 hours (\(a = 0.5\)), and \(X = 2\) hours:

• \(P(x > 2) = \exp(-0.5 \cdot 2) = \exp(-1) \approx 0.3679\).
• \(P(x \leq 2) = 1 - \exp(-0.5 \cdot 2) \approx 0.6321\).
• Mean: \(\mu = 1/0.5 = 2.0000\).
• Median: \(m = \ln(2)/0.5 \approx 1.3863\).
• Variance: \(\sigma^2 = 1/0.5^2 = 4.0000\).
• Standard Deviation: \(\sigma = 1/0.5 = 2.0000\).

5. Frequently Asked Questions (FAQ)

Q: What does the rate parameter represent?
A: The rate parameter \(a\) is the average number of events per unit time, the reciprocal of the mean time between events.

Q: Can the rate or time be negative?
A: No, the rate parameter must be positive, and the time period must be non-negative.

Q: Is the exponential distribution always appropriate?
A: It’s suitable for memoryless processes with constant event rates. For non-constant rates, other distributions may apply.

Q: Why are results in scientific notation?
A: Values less than 0.0001 are displayed in scientific notation for readability.

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