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In many continuous models, like waiting for a car, we always assume the waiting time $t$ to have an exponential distribution. Why is such an assumption appropriate?

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    One possible answer would be to look at http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution2012-05-17
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    There are at least two reasons. First, the exponential distribution appears to be an adequate model for some, maybe many, and perhaps even most, real-life situations, and so the results of the analysis are useful in determining courses of action etc in real life. Second, the model is relatively easy to analyze (and to generalize).2012-05-17
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    Because it [arises](http://en.wikipedia.org/wiki/Poisson_process#Properties) from the [ubiquitous](http://en.wikipedia.org/wiki/Poisson_distribution#How_does_this_distribution_arise.3F_.E2.80.94_The_law_of_rare_events) [Poisson](http://en.wikipedia.org/wiki/Poisson_distribution) [process](http://en.wikipedia.org/wiki/Poisson_process).2012-05-17
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    The [memorylessness](http://en.wikipedia.org/wiki/Memorylessness) property of exponential distribution plays a role too.2012-05-17
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    Because that is the distribution that they will have if the cars' arrival times are independent of one another. If all the cars have left from the same traffic-light-controlled intersection at the same time, then an exponential distribution is not appropriate. But in many cases it is a reasonable assumption.2012-05-17
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    I wish I could find this, but not long ago I was involved in a discussion with someone about modeling the chance that they would get a parking ticket, and I had to persuade them that they should not model the meter readers' arrival times with an exponential distribution, since this would imply that the meter readers operate independent of one another, which is surely unlikely. This is a nice example of a case in which an exponential distribution is *in*appropriate.2012-05-17
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    @MarkDominus Yeah, I think the memorylessness is not a good assumption in some cases, your example is heuristic, thx~2012-05-19
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    @DilipSarwate Got it. Maybe I should deal with some real problem to get a better understand..2012-05-19
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    @bgins Thanks, I'm looking into it.2012-05-19
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    @ArtiomFiodorov Got it, and I think it is the only distribution with memorylessness due to Cauchy functional equation.2012-05-19
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    The memoryless property is a characterization of the exponential disribution. There are a few characterizations for the exponential distribution and the poisson process.2012-05-19
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    [Here's the thing I was thinking about where the meter readers' arrival times were assumed to be exponentially distributed](http://www.reddit.com/r/math/comments/km1wi/method_for_putting_a_coins_in_the_meter/c2lqoik)2012-06-12

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My answer is that we don't. Assuming an exponential arrival time assumes the number of arrivals by time $t$ follows a Poisson process. It is just one of many possible rival time distributions and corresponding point processes that could be used. It is like saying "why do we 'always' assume a normal distribution for continuous random variables?" There too the answer is that we don't. In both cases the method is simple and convenient and there is a limit theorem that sometimes justifies its use.

Recognize that assuming exponential waiting times implies lack of memory. The lack of memory property states that if you are waiting for a bus or car to arrive, and have already waited five minutes, the remaining waiting time has the same exponential distribution that you had when you had just started waiting. This is not always a good assumption. Exponential waiting time/Poisson process are justified under some assumptions of the rarity of events over short time intervals, just like the normal distribution is justified for averages or sums of several observations from some population distribution.