Beta Distribution
The harmonic mean of a beta distribution with shape parameters α and β is:
The harmonic mean with α < 1 is undefined because its defining expression is not bounded in .
Letting α = β
showing that for α = β the harmonic mean ranges from 0 for α = β = 1, to 1/2 for α = β → ∞.
The following are the limits with one parameter finite (non zero) and the other parameter approaching these limits:
With the geometric mean the harmonic mean may be useful in maximum likelihood estimation in the four parameter case.
A second harmonic mean (H1 - X) also exists for this distribution
This harmonic mean with β < 1 is undefined because its defining expression is not bounded in .
Letting α = β in the above expression
showing that for α = β the harmonic mean ranges from 0, for α = β = 1, to 1/2, for α = β → ∞.
The following are the limits with one parameter finite (non zero) and the other approaching these limits:
Although both harmonic means are asymmetric, when α = β the two means are equal.
Read more about this topic: Harmonic Mean
Famous quotes containing the word distribution:
“In this distribution of functions, the scholar is the delegated intellect. In the right state, he is, Man Thinking. In the degenerate state, when the victim of society, he tends to become a mere thinker, or, still worse, the parrot of other mens thinking.”
—Ralph Waldo Emerson (18031882)