MemorylessnessĪn important property of the exponential distribution is that it is memoryless. In accordance with the median-mean inequality. Thus the absolute difference between the mean and median is Where ln refers to the natural logarithm. In light of the examples given above, 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 mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by In general, the reader must check which of these two specifications is being used if an author writes " X ~ Exponential( λ)", since either the notation in the previous (using λ) or the notation in this section (here, using β to avoid confusion) could be intended. Unfortunately this gives rise to a notational ambiguity. This alternative specification is not used here. The alternative specification is sometimes more convenient than the one given above, and some authors will use it as a standard definition.
The parameterisation involving the "rate" parameter arises in the context of events arriving at a rate λ, when the time between events (which might be modelled using an exponential distribution) has a mean of β = λ −1. That is to say, the expected duration of survival of the system is β units of time. In this specification, β is a survival parameter in the sense that if a random variable X is the duration of time that a given biological or mechanical system manages to survive and X ~ Exponential( β) then E = β.
Where β > 0 is a scale parameter of the distribution and is the reciprocal of the rate parameter, λ, defined above. Alternative parameterizationĪ commonly used alternative parameterization is to define the probability density function (pdf) of an exponential distribution as The cumulative distribution function is given byĪlternatively, this can be defined using the Heaviside step function, H( x).
The exponential distribution exhibits infinite divisibility. If a random variable X has this distribution, we write X ~ Exp( λ). The distribution is supported on the interval [0, ∞). Here λ > 0 is the parameter of the distribution, often called the rate parameter. The probability density function (pdf) of an exponential distribution isĪlternatively, this can be defined using the Heaviside step function, H(x).
a process in which events occur continuously and independently at a constant average rate. It describes the time between events in a Poisson process, i.e. negative exponential distribution) is a family of continuous probability distributions. In probability theory and statistics, the exponential distribution (a.k.a. Let $s$ and $t$ be positive, and let's find the conditional probability that the object survives a further $s$ units of time given that it has already survived $t$.Not to be confused with the exponential families of probability distributions. The survival function tells us something unusual about exponentially distributed lifetimes.
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