Inverse Gamma Distribution Mode Calculator

Inverse Gamma Distribution Mode Calculator

For an Inverse Gamma distribution with shape parameter \( \alpha \) and scale parameter \( \beta \), the mode is given by: $$ \text{Mode} = \frac{\beta}{\alpha+1}. $$

* Enter \( \alpha \) (must be \(>0\)) and \( \beta \) (must be \(>0\)).

Step 1: Enter Parameters

e.g., 2

e.g., 3

How It Works

The Inverse Gamma distribution is defined by the PDF: $$ f(x;\alpha,\beta)=\frac{\beta^\alpha}{\Gamma(\alpha)}\,x^{-\alpha-1}e^{-\beta/x},\quad x>0. $$

To find the mode, we differentiate the log of the PDF with respect to \(x\) and set the derivative equal to zero.

This yields the formula: $$ \text{Mode} = \frac{\beta}{\alpha+1}. $$

Simply enter the parameters \(\alpha\) and \(\beta\) to compute the mode.

Formula: \( \text{Mode} = \frac{\beta}{\alpha+1} \)

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