Transformation to Symmetry of Gamma Random Variables

λ

0.344

shape α

2.96

scale β

1.126

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

Z

λ

:

Z

λ

=

( λ X -1)/λ	λ≠0
log(X)	λ=0

Here

λ∈[0,1]

and

X

is a gamma random variable with shape parameter

α

and scale parameter

β

. The mean and variance of

X

are

αβ

and

α

2

β

. So variance stabilization theory suggests that the logarithmic transformation should be used. This Demonstration shows that the value of

λ

that works best for making the distribution of

Z

λ

symmetric depends on the shape

α

. The case of

α=1

with

λ=0.3

shown in the thumbnail works fairly well. With

λ=0.3

, try experimenting with other shape parameters

α

to see how well this transformation preserves symmetry. Compare how well the logarithmic transformation,

λ=0

, works to make the distribution symmetric. With

λ=1

fixed, explore the shapes of various gamma distributions. Note that as

α

increases, the gamma distribution approximates the normal distribution. Changing the scale parameter

β

enlarges or shrinks the distribution but has no effect on the skewness or lack of symmetry.