Efficiently compute Inverse Gamma distribution parameters—mean, variance, mode, PDF, and more—with accurate online calculators ideal for Bayesian statistics and reliability engineering.

Inverse Gamma Distribution Calculator

For parameters $\alpha>0$ and $\beta>0$, the PDF is:

$$ f(x;\alpha,\beta)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{-\alpha-1}e^{-\beta/x},\quad x>0. $$

and the CDF is:

$$ F(x;\alpha,\beta)=1-P\Bigl(\alpha,\frac{\beta}{x}\Bigr)=\frac{\Gamma\Bigl(\alpha,\frac{\beta}{x}\Bigr)}{\Gamma(\alpha)},\quad x>0, $$

where $P(\alpha,\beta/x)$ is the lower regularized gamma function.

Step 1: Enter Parameters

$\alpha$ (shape):

Enter a positive number (e.g., 3)

$\beta$ (scale):

Enter a positive number (e.g., 2)

$ x $ (value to evaluate, $x>0$):

Enter a number greater than 0 (e.g., 1)

Understanding the Inverse Gamma Distribution

The Inverse Gamma distribution is a continuous probability distribution frequently utilized in Bayesian statistics, especially as a prior distribution for variance parameters. It plays a significant role in various analytical and predictive modeling scenarios across multiple disciplines, including reliability engineering and risk assessment.

The Inverse Gamma distribution is characterized by two key parameters:

Shape Parameter (α): Influences the distribution's form, affecting its skewness and peak.
Scale Parameter (β): Adjusts the distribution's scale or dispersion.

Key Applications of the Inverse Gamma Distribution

This distribution is particularly beneficial in:

Bayesian Statistics: Commonly used as a prior distribution in hierarchical and Bayesian models for variance parameters.
Reliability Engineering: Modeling lifetimes and failure times of components and systems.
Risk Assessment: Useful in fields requiring precise modeling of uncertain or variable outcomes.