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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, an inverse distribution is the distribution of the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of a random variable. Inverse distributions arise in particular in the
Bayesian Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a followe ...
context of
prior distribution In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...
s and
posterior distribution The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
s for
scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...
s. In the
algebra of random variables The algebra of random variables in statistics, provides rules for the symbolic manipulation of random variables, while avoiding delving too deeply into the mathematically sophisticated ideas of probability theory. Its symbolism allows the treat ...
, inverse distributions are special cases of the class of
ratio distribution A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually independent) random variables ''X'' ...
s, in which the numerator random variable has a
degenerate distribution In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter d ...
.


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 ...
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 g ...
with
density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
''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. Eve ...
''F''(''x''), then the cumulative distribution function, ''G''(''y''), of the reciprocal is found by noting that : G(y) = \Pr(Y \leq y) = \Pr\left(X \geq \frac\right) = 1-\Pr\left(X<\frac\right) = 1 - F\left( \frac \right). Then the density function of ''Y'' is found as the derivative of the cumulative distribution function: : g(y) = \frac f\left( \frac \right) .


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
:f(x) \propto x^ \quad \text 0 where \propto \!\, means "is proportional to". It follows that the inverse distribution in this case is of the form :g(y) \propto y^ \quad \text 0\le b^ which is again a reciprocal distribution.


Inverse uniform distribution

If the original random variable ''X'' is uniformly distributed on the interval (''a'',''b''), where ''a''>0, then the reciprocal variable ''Y'' = 1 / ''X'' has the reciprocal distribution which takes values in the range (''b''−1 ,''a''−1), and the probability density function in this range is : g( y ) = y^ \frac , and is zero elsewhere. The cumulative distribution function of the reciprocal, within the same range, is : G( y ) = \frac . For example, if ''X'' is uniformly distributed on the interval (0,1), then ''Y'' = 1 / ''X'' has density g( y ) = y^ and cumulative distribution function G( y ) = when y > 1 .


Inverse ''t'' distribution

Let ''X'' be a ''t'' distributed random variate with ''k''
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
. Then its density function is : f( x ) = \frac \frac \frac . The density of ''Y'' = 1 / ''X'' is : g( y ) = \frac \frac \frac . With ''k'' = 1, the distributions of ''X'' and 1 / ''X'' are identical (''X'' is then Cauchy distributed (0,1)). If ''k'' > 1 then the distribution of 1 / ''X'' is
bimodal In statistics, a multimodal distribution is a probability distribution with more than one mode. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and d ...
.


Reciprocal normal distribution

If variable ''X'' follows a
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
\mathcal(\mu,\sigma^2), then the inverse or reciprocal ''Y''=1/''X'' follows a reciprocal normal distribution: : f(y) = \frac e^ . If variable ''X'' follows a
standard normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
\mathcal(0, 1), then ''Y''=1/''X'' follows a ''reciprocal standard normal distribution'',
heavy-tailed In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distrib ...
and
bimodal In statistics, a multimodal distribution is a probability distribution with more than one mode. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and d ...
, with modes at \pm\tfrac and density f(y)=\frac and the first and higher-order moments do not exist. For such inverse distributions and for
ratio distribution A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually independent) random variables ''X'' ...
s, there can still be defined probabilities for intervals, which can be computed either by
Monte Carlo simulation Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...
or, in some cases, by using the Geary–Hinkley transformation. However, in the more general case of a shifted reciprocal function 1/(p-B), for B=N(\mu,\sigma) following a general normal distribution, then mean and variance statistics do exist in a
principal value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a posit ...
sense, if the difference between the pole p and the mean \mu is real-valued. The mean of this transformed random variable (''reciprocal shifted normal distribution'') is then indeed the scaled
Dawson's function In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function. Definition The Dawson function is defined as either: D_+(x) = e^ \int_0^x e^\,dt, a ...
: :\frac F \left(\frac\right). In contrast, if the shift p-\mu is purely complex, the mean exists and is a scaled
Faddeeva function The Faddeeva function or Kramp function is a scaled complex complementary error function, :w(z):=e^\operatorname(-iz) = \operatorname(-iz) =e^\left(1+\frac\int_0^z e^\textt\right). It is related to the Fresnel integral, to Dawson's integral, an ...
, whose exact expression depends on the sign of the imaginary part, \operatorname(p-\mu). In both cases, the variance is a simple function of the mean. Therefore, the variance has to be considered in a principal value sense if p-\mu is real, while it exists if the imaginary part of p-\mu is non-zero. Note that these means and variances are exact, as they do not recur to linearisation of the ratio. The exact covariance of two ratios with a pair of different poles p_1 and p_2 is similarly available. The case of the inverse of a complex normal variable B, shifted or not, exhibits different characteristics.


Inverse exponential distribution

If X is an exponentially distributed random variable with rate parameter \lambda, then Y=1/X has the following cumulative distribution function: F_Y(y) = e^for y> 0. Note that the expected value of this random variable does not exist. The reciprocal exponential distribution finds use in the analysis of fading wireless communication systems.


Inverse Cauchy distribution

If ''X'' is a Cauchy distributed (''μ'', ''σ'') random variable, then 1 / X is a Cauchy ( ''μ'' / ''C'', ''σ'' / ''C'' ) random variable where ''C'' = ''μ''2 + ''σ''2.


Inverse F distribution

If ''X'' is an ''F''(''ν''1, ''ν''2 ) distributed random variable then 1 / ''X'' is an ''F''(''ν''2, ''ν''1 ) random variable.


Reciprocal of binomial distribution

No closed form for this distribution is known. An asymptotic approximation for the mean is known.Cribari-Neto F, Lopes Garcia N, Vasconcellos KLP (2000) A note on inverse moments of binomial variates. Brazilian Review of Econometrics 20 (2) E ( 1 + X )^a = O( ( np )^ ) + o( n^ ) where E[] is the expectation operator, X is a random variable, O() and o() are the big and little Big O notation, o order functions, n is the sample size, p is the probability of success and a is a variable that may be positive or negative, integer or fractional.


Reciprocal of triangular distribution

For a
triangular distribution In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit ''a'', upper limit ''b'' and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''. ...
with lower limit ''a'', upper limit ''b'' and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b'', the mean of the reciprocal is given by \mu = \frac and the variance by \sigma^2 = \frac - \mu^2. Both moments of the reciprocal are only defined when the triangle does not cross zero, i.e. when ''a'', ''b'', and ''c'' are either all positive or all negative.


Other inverse distributions

Other inverse distributions include
:
inverse-chi-squared distribution In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) ''Bayesian Theory'' ,Wiley (pages 119, 431) ) is a continuous probability distribution of a positive-val ...
:
inverse-gamma distribution In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
: inverse-Wishart distribution : inverse matrix gamma distribution


Applications

Inverse distributions are widely used as prior distributions in Bayesian inference for scale parameters.


See also

*
Harmonic mean In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the recipro ...
*
Ratio distribution A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually independent) random variables ''X'' ...
* Self-reciprocal distributions


References

{{reflist Algebra of random variables Types of probability distributions