the mean of the exponential distribution is
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Among all continuous probability distributions with support [0, ∞) and mean μ, the exponential distribution with λ = 1/μ has the largest differential entropy. In other words, it is the maximum entropy probability distribution for a random variate X which is greater than or equal to zero and for which E[X] is fixed.
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The probability distribution of the time between events in a poisson point process is called exponential distribution.
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