# Transformation to Symmetry of Gamma Random Variables

Transformation to Symmetry of Gamma Random Variables

This Demonstration shows the probability density function for the transformed random variable, :

Z

λ

Z

λ

( λ X | λ≠0 |

log(X) | λ=0 |

Here and is a gamma random variable with shape parameter and scale parameter . The mean and variance of are and . So variance stabilization theory suggests that the logarithmic transformation should be used. This Demonstration shows that the value of that works best for making the distribution of symmetric depends on the shape . The case of with shown in the thumbnail works fairly well. With , try experimenting with other shape parameters to see how well this transformation preserves symmetry. Compare how well the logarithmic transformation, , works to make the distribution symmetric. With fixed, explore the shapes of various gamma distributions. Note that as increases, the gamma distribution approximates the normal distribution. Changing the scale parameter enlarges or shrinks the distribution but has no effect on the skewness or lack of symmetry.

λ∈[0,1]

X

α

β

X

αβ

α

2

β

λ

Z

λ

α

α=1

λ=0.3

λ=0.3

α

λ=0

λ=1

α

β