April 2015 Archives

A Costly Proposition

Now that we know the statistical properties of memoryless processes, being those in which the waiting time for the occurence of an event is independent of how long we have already been waiting for it, I think it would make for an interesting postscript to briefly cover how we might use them to model real world problems.

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The Longest Wait

We have seen how the waiting time for an event in a memoryless process, being one in which the probability that we must wait some given time doesn't change no matter how long we've already been waiting, must be exponentially distributed, that the waiting time for the kth such event must be gamma distributed and that the number of such events occurring in one unit of time must be Poisson distributed.
This time I'd like to ask how long we must wait for the first and the last of several such processes that are happening at the same time to trigger an event.

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...And Then Three Came Along At Once!

In the last few posts we have been looking at the properties of memoryless processes, being those processes that trigger events at intervals that are independent of how long we have been waiting for one to happen. First we answered the question of what is the probability that we must wait some given time that an event will occur with the exponential distribution, and then the question of what is the probability that we must wait some given time for the kth event to occur with the gamma distribution, which allows the somewhat counter-intuitive case of non-integer k.

This time I'd like to ask another question; what is the probability that we'll observe k events, occurring at a rate λ, in a period of one unit of time?

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