Inverse Gamma Distribution Inverse CDF Calculator

Inverse Gamma Distribution Inverse CDF Calculator

For an Inverse Gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\), the CDF is $$ F(x;\alpha,\beta)=1-P\left(\alpha,\frac{\beta}{x}\right), $$ where \(P(\alpha,z)\) is the regularized lower incomplete gamma function.

This calculator finds the quantile \( x \) such that \( F(x;\alpha,\beta)=q \).

* Enter a probability \( q \) (0 < \( q \) < 1), \(\alpha>0\), and \(\beta>0\).

Step 1: Enter Parameters

Enter a probability between 0 and 1 (e.g., 0.5).

e.g., 2

e.g., 3

How It Works

The CDF of an Inverse Gamma distribution is given by:

$$ F(x;\alpha,\beta)=\frac{\Gamma\left(\alpha,\frac{\beta}{x}\right)}{\Gamma(\alpha)} = 1-P\left(\alpha,\frac{\beta}{x}\right), $$

where \(P(\alpha,z)\) is the regularized lower incomplete gamma function.

The calculator uses a bisection method to find the value \( x \) such that \( F(x;\alpha,\beta)=q \), i.e., it solves for \( x \) satisfying:

$$ 1-P\left(\alpha,\frac{\beta}{x}\right)=q. $$

Note: Adjust parameters as needed. For proper results, ensure 0 < \( q \) < 1, \( \alpha>0 \), and \( \beta>0 \).