1. What is Gamma Distribution?

The Gamma distribution is a two-parameter family of continuous probability distributions. It is frequently used to model waiting times, reliability data, and other positive-valued variables in statistics and engineering.

2. How Does the Calculator Work?

The calculator uses the Gamma probability density function (PDF):

Where:

• \( x \) — The value at which to evaluate the function (x > 0)
• \( k \) — Shape parameter (k > 0)
• \( \theta \) — Scale parameter (θ > 0)
• \( \Gamma(k) \) — Gamma function evaluated at k

Explanation: The Gamma distribution generalizes several other distributions including exponential and chi-squared distributions.

3. Importance of Gamma Distribution

Details: Gamma distribution is widely used in Bayesian statistics, queuing models, climatology, and financial modeling. It's particularly useful for modeling right-skewed continuous data.

4. Using the Calculator

Tips: Enter positive values for x, shape (k), and scale (θ). The calculator will compute the probability density at point x.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between Gamma and exponential distribution?
A: Exponential distribution is a special case of Gamma distribution when shape parameter k = 1.

Q2: How is Gamma distribution related to chi-squared?
A: Chi-squared distribution with ν degrees of freedom is Gamma(k=ν/2, θ=2).

Q3: What are typical applications of Gamma distribution?
A: Rainfall modeling, insurance claims modeling, reliability engineering, and Bayesian conjugate priors.

Q4: What's the difference between shape and scale parameters?
A: Shape (k) affects the form of the distribution, while scale (θ) stretches or compresses it.

Q5: Can Gamma distribution model left-skewed data?
A: No, Gamma distribution is only right-skewed. For left-skewed data, you might consider inverse Gamma.