If I have a time series that is usually $0$, but occasionally peaks at $1$ (what I'll term an event). If I know the time between 'events', is there a way to define an event probability?
Probability of sporadic events
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probability
1 Answers
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Yes, if you actually know the underlying distribution of time in between events. A very common distribution for discrete events is the Poisson distribution, where the time in between events (waiting time) is independent of any other waiting time and is distributed as a exponential random variables, i.e., the waiting times are distributed as $\sim e^{-t}.$
This results in the Poisson distribution which tells you the probability of $n$ events happening in some interval that has $\lambda$ events on average:
$$p(n)=\frac{\lambda^ne^{-\lambda}}{n!}$$
This was, however, just an example. If you do not know the underlying distribution of waiting times exactly, you need to make a model that fits your particular data set.
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0Thanks! If I get a histogram of the waiting time I should be able to see if it's $e^{-t}$ or not. – 2017-01-18
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0@James Yes, with a constant $re^{-rt}$, see https://en.wikipedia.org/wiki/Exponential_distribution. – 2017-01-18