distribution) The Hoyt distribution can be obtained from exponential distribution and arcsine distribution The exponential distribution is a limit of the κ-exponential distribution in the case. Exponential distribution is a limit of the κ-Generalized Gamma distribution in the and cases: Other related distributions: Hyper-exponential distribution – the distribution whose density is a weighted sum of exponential densities. Hypoexponential distribution – the distribution of a general sum of exponential random variables. exGaussian distribution – the sum of an exponential distribution and a normal distribution. Statistical inference Below, suppose random variable X is exponentially distributed with rate parameter λ, and are n independent samples from X, with sample mean . Parameter estimation The maximum likelihood estimator for λ is constructed as follows. The likelihood function for λ, given an independent and identically distributed sample x = (x1, …, xn) drawn from the variable, is: where: is the sample mean. The derivative of the likelihood function's logarithm is: Consequently, the maximum likelihood estimate for the rate parameter is: This is an unbiased estimator of although an unbiased MLE estimator of and the distribution mean. The bias of is equal to which yields the bias-corrected maximum likelihood estimator An approximate minimizer of mean squared error (see also: bias–variance tradeoff) can be found, assuming a sample size greater than two, with a correction factor to the MLE: This is derived from the mean and variance of the inverse-gamma distribution, . Fisher information The Fisher information, denoted , for an estimator of the rate parameter is given as: Plugging in the distribution and solving gives: This determines the amount of information each independent sample of an exponential distribution carries about the unknown rate parameter . Confidence intervals The 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by: which is also equal to: where is the percentile of the chi squared distribution with v degrees of freedom, n is the number of observations of inter-arrival times in the sample, and x-bar is the sample average. A simple approximation to the exact interval endpoints can be derived using a normal approximation to the distribution. This approximation gives the following values for a 95% confidence interval: This approximation may be acceptable for samples containing at least 15 to 20 elements. Bayesian inference The conjugate prior for the exponential distribution is the gamma distribution (of which the exponential distribution is a special case).