event over some initial period of time s, the distribution of the remaining waiting time is the same as the original unconditional distribution. For example, if an event has not occurred after 30 seconds, the conditional probability that occurrence will take at least 10 more seconds is equal to the unconditional probability of observing the event more than 10 seconds after the initial time. The exponential distribution and the geometric distribution are the only memoryless probability distributions. The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant failure rate. Quantiles The quantile function (inverse cumulative distribution function) for Exp(λ) is The quartiles are therefore: first quartile: ln(4/3)/λ median: ln(2)/λ third quartile: ln(4)/λ And as a consequence the interquartile range is ln(3)/λ. Conditional Value at Risk (Expected Shortfall) The conditional value at risk (CVaR) also known as the expected shortfall or superquantile for Exp(λ) is derived as follows : Buffered Probability of Exceedance (bPOE) The buffered probability of exceedance is one minus the probability level at which the CVaR equals the threshold . It is derived as follows : Kullback–Leibler divergence The directed Kullback–Leibler divergence in nats of ("approximating" distribution) from ('true' distribution) is given by Maximum entropy distribution Among all continuous probability distributions with support 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. Distribution of the minimum of exponential random variables Let X1, …, Xn be independent exponentially distributed random variables with rate parameters λ1, …, λn. Then is also exponentially distributed, with parameter This can be seen by considering the complementary cumulative distribution function: The index of the variable which achieves the minimum is distributed according to the categorical distribution A proof can be seen by letting . Then, Note that is not exponentially distributed, if X1, …, Xn do not all have parameter 0. Joint moments of i.i.d. exponential order statistics Let be independent and identically distributed exponential random variables with rate parameter λ. Let denote the corresponding order statistics. For , the joint moment of the order statistics and is given by This can be seen by invoking the law of total expectation and the memoryless property: The first equation follows from the law of