obtained from the distribution of the pivotal quantity ; a profile predictive likelihood, obtained by eliminating the parameter λ from the joint likelihood of xn+1 and λ by maximization; an objective Bayesian predictive posterior distribution, obtained using the non-informative Jeffreys prior 1/λ; the Conditional Normalized Maximum Likelihood (CNML) predictive distribution, from information theoretic considerations. The accuracy of a predictive distribution may be measured using the distance or divergence between the true exponential distribution with rate parameter, λ0, and the predictive distribution based on the sample x. The Kullback–Leibler divergence is a commonly used, parameterisation free measure of the difference between two distributions. Letting Δ(λ0||p) denote the Kullback–Leibler divergence between an exponential with rate parameter λ0 and a predictive distribution p it can be shown that where the expectation is taken with respect to the exponential distribution with rate parameter , and is the digamma function. It is clear that the CNML predictive distribution is strictly superior to the maximum likelihood plug-in distribution in terms of average Kullback–Leibler divergence for all sample sizes . Random variate generation A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval , the variate has an exponential distribution, where F is the quantile function, defined by Moreover, if U is uniform on (0, 1), then so is 1 − U. This means one can generate exponential variates as follows: Other methods for generating exponential variates are discussed by Knuth and Devroye. A fast method for generating a set of ready-ordered exponential variates without using a sorting routine is also available. See also Dead time – an application of exponential distribution to particle detector analysis. Laplace distribution, or the "double exponential distribution". Relationships among probability distributions Marshall–Olkin exponential distribution References External links Online calculator of Exponential Distribution Continuous distributions Exponentials Poisson point processes Conjugate prior distributions Exponential family distributions Infinitely divisible probability distributions Survival analysis