page_title: Exponential distribution
section_title: Mean, variance, moments and median
hierarchical_section_title: Exponential distribution / Properties / Mean, variance, moments and median
context_page_description: In probability theory and statistics, the 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, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others.
context_section_description: 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.
caption: The median is the preimage F−1(1/2).