In
probability theory and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, an inverse distribution is the distribution of the
reciprocal of a random variable. Inverse distributions arise in particular in the
Bayesian context of
prior distributions and
posterior distributions for
scale parameters. In the
algebra of random variables, inverse distributions are special cases of the class of
ratio distributions, in which the numerator random variable has a
degenerate distribution.
Relation to original distribution
In general, given the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
of a random variable ''X'' with strictly positive support, it is possible to find the distribution of the reciprocal, ''Y'' = 1 / ''X''. If the distribution of ''X'' is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
with
density function ''f''(''x'') and
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
''F''(''x''), then the cumulative distribution function, ''G''(''y''), of the reciprocal is found by noting that
:
Then the density function of ''Y'' is found as the derivative of the cumulative distribution function:
:
Examples
Reciprocal distribution
The
reciprocal distribution has a density function of the form.
[ Hamming R. W. (1970]
"On the distribution of numbers"
''The Bell System Technical Journal'' 49(8) 1609–1625
: