Inverted gamma distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ inverted gamma(α,β) is used to indicate that the random variable X has the inverted gamma distribution with positive parameters α and β. An inverted gamma random variable X with shape parameter α and scale parameter β has probability density function

f(x) = \frac{x^{−(α+1)} e^{−1/(βx)}}{Γ(α)β^{α}} x > 0.

The probability density function with two different parameter settings is illustrated below.

PSfrag replacements α = 1, β = 1

α = 1.5, β = 1
x
f(x)

The mode of X is

\frac{β}{α + 1} .

The population mean, variance, skewness, and kurtosis of X are

E[X] = \frac{1}{β(α − 1)} , α > 1 V [X] = \frac{1}{β^{2}(α − 1)^{2}(α − 2)} , α > 2

E [( \frac{X − μ}{σ} ) ^{3}] = \frac{4}{(α − 1)(α − 3)\sqrt{1/(α − 2)(α − 1)^{2}}} , α > 3

E [( \frac{X − μ}{σ} ) ^{4}] = \frac{3(α − 2))(α + 5)}{(α − 3)(α − 4)} , α > 4.

APPL verification: The APPL statements

X := InvertedGammaRV(alpha, beta); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the population mean, variance, skewness, and kurtosis.