Inverse Gamma Distribution Variance Calculator

Inverse Gamma Distribution Variance Calculator

For an Inverse Gamma distribution with shape parameter \( \alpha \) and scale parameter \( \beta \), the variance is given by: $$ \text{Variance} = \frac{\beta^2}{(\alpha-1)^2(\alpha-2)}, \quad \text{for } \alpha > 2. $$

* Enter \( \alpha \) (must be \(>0\)) and \( \beta \) (must be \(>0\)). Note: The variance is defined only if \(\alpha > 2\).

Step 1: Enter Parameters

e.g., 3 (must be > 2 for variance to be defined)

e.g., 4

How It Works

The Inverse Gamma distribution is defined by the probability density function:

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

The variance of this distribution is given by:

$$ \text{Variance} = \frac{\beta^2}{(\alpha-1)^2(\alpha-2)} \quad \text{for } \alpha > 2. $$

If \(\alpha \le 2\), the variance is undefined (or infinite).

Formula: \( \text{Variance} = \frac{\beta^2}{(\alpha-1)^2(\alpha-2)} \) (valid for \(\alpha > 2\))

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