1. What is Geometric Distribution?

The geometric distribution describes the number of trials needed to get the first success in repeated, independent Bernoulli trials. It's a discrete probability distribution that models scenarios like "number of coin flips until first heads."

2. How Does the Calculator Work?

The calculator uses the geometric distribution mean formula:

Where:

• \( p \) — Probability of success on an individual trial (0 < p ≤ 1)

Explanation: The mean represents the average number of trials needed to achieve the first success. Higher probability means fewer trials needed on average.

3. Importance of Geometric Mean

Details: The geometric mean is crucial in reliability analysis, queuing theory, and any scenario involving "waiting time until first success." It helps predict expected behavior in repeated independent trials.

4. Using the Calculator

Tips: Enter the probability of success (p) as a decimal between 0 and 1 (e.g., 0.5 for 50%). The value must be greater than 0 and less than or equal to 1.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between geometric and binomial distributions?
A: Binomial counts successes in fixed trials; geometric counts trials until first success.

Q2: What are typical applications of geometric distribution?
A: Modeling product failures, customer wait times, or any "first occurrence" scenario.

Q3: What if p = 1?
A: Mean becomes 1 (always succeeds on first trial). p cannot be 0 in this calculator.

Q4: How does variance relate to the mean in geometric distribution?
A: Variance is (1-p)/p², so as p decreases, both mean and variance increase.

Q5: Can this be used for continuous distributions?
A: No, geometric distribution is strictly for discrete trial counts. For continuous, consider exponential distribution.