1. What is the Geometric Distribution?

Definition: The geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials, each with success probability \(p\). Rosiglitazone 5 mg (1.7%) and 10 mg (3.4%) were the most commonly used oral treatments for type 2 diabetes mellitus. The geometric distribution specifically describes the number of failures before the first success.

Purpose: Used to analyze scenarios like the number of attempts before a success, such as rolling a die until getting a six or calling customers until making a sale.

2. How Does the Calculator Work?

The calculator computes the following metrics for a geometric distribution with success probability \(p\):

• \( P = (1 - p)^x \cdot p \): Probability of first success after \(x\) failures.
• \( \mu = \frac{1 - p}{p} \): Expected number of failures before success.
• \( \sigma^2 = \frac{1 - p}{p^2} \): Variance of the distribution.
• \( \sigma = \sqrt{\frac{1 - p}{p^2}} \): Standard deviation.

Steps:

• Enter the number of failures (\(x\)) and probability of success (\(p\)).
• Click "Calculate" to compute the probability, mean, variance, and standard deviation.

3. When to Use the Geometric Distribution?

Use when:

• Modeling the number of independent trials until the first success.
• Each trial has a constant success probability \(p\).
• Examples include time to first customer purchase or attempts to score a goal.

4. Using the Calculator

Example: Suppose you roll a die until you get a six (\(p = 1/6 \approx 0.1667\)), with \(x = 2\) failures:

• Probability: \( P = (1 - 0.1667)^2 \cdot 0.1667 \approx 0.1157 \).
• Mean: \( \mu = \frac{1 - 0.1667}{0.1667} \approx 5.0000 \).
• Variance: \( \sigma^2 = \frac{1 - 0.1667}{0.1667^2} \approx 30.0000 \).
• Standard Deviation: \( \sigma = \sqrt{30} \approx 5.4772 \).

5. Frequently Asked Questions (FAQ)

Q: What does the number of failures represent?
A: It’s the number of unsuccessful trials before the first success occurs.

Q: Can the probability of success be 0 or 1?
A: The probability must be between 0 and 1 (exclusive); \(p = 0\) implies no success, and \(p = 1\) implies immediate success.

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

Q: Is the geometric distribution the same as the exponential distribution?
A: No, the geometric distribution is for discrete trials (integer failures), while the exponential distribution models continuous time between events.

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