1. What is Geometric Distribution Variance?

The geometric distribution variance measures the spread in the number of trials needed to get the first success in repeated Bernoulli trials. It's a fundamental concept in probability theory and statistics.

2. How Does the Calculator Work?

The calculator uses the geometric variance formula:

Where:

• \( p \) — Probability of success on an individual trial (0 < p ≤ 1)
• \( \text{var} \) — Variance of the geometric distribution

Explanation: The variance increases as the probability of success decreases, showing greater uncertainty when success is rare.

3. Importance of Geometric Variance

Details: Understanding the variance helps in predicting the reliability of the expected number of trials needed for the first success. It's used in quality control, reliability engineering, and risk assessment.

4. Using the Calculator

Tips: Enter the probability of success (p) as a value between 0 and 1. For example, for a 30% chance, enter 0.3.

5. Frequently Asked Questions (FAQ)

Q1: What's the range of possible variance values?
A: Variance ranges from 0 (when p approaches 1) to infinity (as p approaches 0).

Q2: How is this different from binomial variance?
A: Binomial variance is for number of successes in n trials, while geometric variance is for number of trials until first success.

Q3: What are typical applications?
A: Used in modeling scenarios like number of attempts until first login success, or components tested until first failure.

Q4: What if p = 0 or p = 1?
A: p=0 is undefined (no success possible), p=1 gives variance=0 (always succeed on first try).

Q5: What's the relationship to the mean?
A: The mean is 1/p, and variance is (1-p)/p², so variance = mean × (mean - 1) when p ≤ 1.