Reciprocal distribution

Reciprocal

Parameters	$0 < a < b, a, b \in \R$
Support	$[ a , b ]$
PDF	$\frac{ 1 }{ x [ \log_e( b ) - \log_e( a ) ] }$
CDF	$\frac{ \log_e( x ) - \log_e( a ) }{ \log_e(b) - \log_e( a ) }$
Mean	$\frac{ b - a }{ \log_e( b ) - \log_e( a ) }$

In probability and statistics, the reciprocal distribution is a continuous probability distribution. It is characterised by its probability density function, within the support of the distribution, being proportional to the reciprocal of the variable.

The reciprocal distribution is an example of an inverse distribution, and the reciprocal (inverse) of a random variable with a reciprocal distribution itself has a reciprocal distribution.

Definition

The probability density function (pdf) of the reciprocal distribution is

$f( x; a,b ) = \frac{ 1 }{ x [ \log_e( b ) - \log_e( a ) ]} \quad \text{ for } a \le x \le b \text{ and } a > 0.$

Here, $a$ and $b$ are the parameters of the distribution, which are the lower and upper bounds of the support, and $\log_e$ is the natural log function (the logarithm to base e). The cumulative distribution function is

$F( x ; a,b) = \frac{ \log_e( x ) - \log_e( a ) }{ \log_e( b ) - \log_e( a ) } \quad \text{ for } a \le x \le b.$

Differential equation

The pdf of the reciprocal distribution is a solution to the following differential equation:

$\left\{\begin{array}{l} x f'(x)+f(x)=0, \\ f(1)=\frac{1}{\log (b)-\log (a)} \end{array}\right\}$

Applications

The reciprocal distribution is of considerable importance in numerical analysis as a computer’s arithmetic operations transform mantissas with initial arbitrary distributions to the reciprocal distribution as a limiting distribution.^[1]

References

1. Hamming R. W. (1970) "On the distribution of numbers", The Bell System Technical Journal 49(8) 1609–1625