In general, given the
probability distribution 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 with
density functionf(x) and
cumulative distribution functionF(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:
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
and is zero elsewhere.
The cumulative distribution function of the reciprocal, within the same range, is
For example, if X is uniformly distributed on the interval (0,1), then Y = 1 / X has density and cumulative distribution function when
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.[citation needed]
and the first and higher-order moments do not exist.[2] For such inverse distributions and for
ratio distributions, there can still be defined probabilities for intervals, which can be computed either by
Monte Carlo simulation or, in some cases, by using the Geary–Hinkley transformation.[3]
However, in the more general case of a shifted reciprocal function , for following a general normal distribution, then mean and variance statistics do exist in a
principal value sense, if the difference between the pole and the mean is real-valued. The mean of this transformed random variable (reciprocal shifted normal distribution) is then indeed the scaled
Dawson's function:[4]
In contrast, if the shift is purely complex, the mean exists and is a scaled
Faddeeva function, whose exact expression depends on the sign of the imaginary part, .
In both cases, the variance is a simple function of the mean.[5] Therefore, the variance has to be considered in a principal value sense if is real, while it exists if the imaginary part of 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 and is similarly available.[6]
The case of the inverse of a
complex normal variable, shifted or not, exhibits different characteristics.[4]
Inverse exponential distribution
If is an exponentially distributed random variable with rate parameter , then has the following cumulative distribution function: for . 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 = μ2 + σ2.
Inverse F distribution
If X is an
F(ν1, ν2 ) distributed random variable then 1 / X is an F(ν2, ν1 ) random variable.
Reciprocal of binomial distribution
If is distributed according to a Binomial distribution with number of trials and a probability of success then no closed form for the reciprocal distribution is known. However, we can calculate the mean of this distribution.
An asymptotic approximation for the non-central moments of the reciprocal distribution is known.[7]
where O() and o() are the big and little
o order functions and is a real number.
Reciprocal of triangular distribution
For a
triangular distribution 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
and the variance by
.
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.
^
abLecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems". Journal of Sound and Vibration. 332 (11): 2750–2776.
doi:
10.1016/j.jsv.2012.12.009.
^Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems". Journal of Sound and Vibration. 332 (11). Section (4.1.1).
doi:
10.1016/j.jsv.2012.12.009.
^Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems". Journal of Sound and Vibration. 332 (11). Eq.(39)-(40).
doi:
10.1016/j.jsv.2012.12.009.
^Cribari-Neto F, Lopes Garcia N, Vasconcellos KLP (2000) A note on inverse moments of binomial variates. Brazilian Review of Econometrics 20 (2)