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Concept
Exponential distribution
Summary
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions.
The probability density function (pdf) of an exponential distribution is
Here λ > 0 is the parameter of the distribution, often called the rate parameter. The distribution is supported on the interval . If a random variable X has this distribution, we write X ~ Exp(λ).
The exponential distribution exhibits infinite divisibility.
The cumulative distribution function is given by
The exponential distribution is sometimes parametrized in terms of the scale parameter β = 1/λ, which is also the mean:
The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by
In light of the examples given below, this makes sense: if you receive phone calls at an average rate of 2 per hour, then you can expect to wait half an hour for every call.
The variance of X is given by
so the standard deviation is equal to the mean.
The moments of X, for are given by
The central moments of X, for are given by
where !n is the subfactorial of n
The median of X is given by
where ln refers to the natural logarithm. Thus the absolute difference between the mean and median is
in accordance with the median-mean inequality.
Official source
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