If the inter-arrival time between consecutive events follow an exponential distribution with mean 1/$\lambda$ the number of events over a time period t can be calculated based on Poisson distribution with mean $\lambda$t. Based on this finding, is it possible to calculate P(n<7) using an Exponential distribution?
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1$\begingroup$ When you say "using an exponential distribution" do you mean to do something other than take advantage of the resulting Poisson distribution or do you mean using that relationship to the Poisson? Is the time in which these $n$ events are to occur known? $\endgroup$– Glen_bCommented Sep 23, 2016 at 0:46
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$\begingroup$ I don't understand the question. You just said that the number of events in a fixed interval is Poisson distributed. Why would you then calculate probabilities with an exponential distribution? You can of course, it is the probability that a sum of 7 iid exponential random variables is greater than $\lambda t $, but this requires an integral of the gamma distribution which will just give another expression for the same result. $\endgroup$– IanCommented Nov 2, 2016 at 19:05
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It is possible but not simple - you need to find the distribution of sums of Exponential random variables which will result in Gamma distributions.
