The reciprocal distribution has a density function of the form. where means "is proportional to". It follows that the inverse distribution in this case is of the form which is again a reciprocal distribution.
If the original random variable X is uniformly distributed on the interval, where a>0, then the reciprocal variable Y = 1 / X has the reciprocal distribution which takes values in the range, 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
Inverse ''t'' distribution
Let X be a t distributed random variate with k degrees of freedom. Then its density function is The density of Y = 1 / X is With k = 1, the distributions of X and 1 / X are identical. If k > 1 then the distribution of 1 / X is bimodal.
Reciprocal normal distribution
If X is a standard normally distributed variable then the distribution of the inverse or reciprocal 1/X is bimodal, and the first and higher-order moments do not exist. 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. 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 is then indeed the scaled Dawson's function: 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. 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. The case of the inverse of a complex normal variable, shifted or not, exhibits different characteristics.
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.
No closed form for this distribution is known. An asymptotic approximation for the mean is known. where E is the expectation operator, X is a random variable, O and o are the big and little 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.
For a triangular distribution with lower limita, upper limitb 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.
Other inverse distributions
Other inverse distributions include
Applications
Inverse distributions are widely used as prior distributions in Bayesian inference for scale parameters.
