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