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From Poisson's postulates, we know Poisson works for rare events. However, we also know binomial is an approximation of Poisson when the probability of an event is small. So can we use binomial and Poisson interchangeably for rare event? What is the benefit of using one rather than the other?

user3119750
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1 Answers1

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More typically, the Poisson is used to approximate the binomial for situations when $p$ is small but $n$ is "large enough" for the approximation to be reasonable rather than the binomial being used to approximate the Poisson.

With large $n$, Poisson probabilities are easier to calculate. Further, the binomial requires a table for each $n,p$ combination, while the Poisson only requires one for each $\lambda$.

Another advantage of using the Poisson: when looking at the distribution of the sum of two (or more) binomials with different $p$ -- each of which is well approximated by a Poisson -- the sum won't be binomial, but will still be approximately Poisson.

You could sometimes use a binomial to approximate a Poisson, but there would usually be little reason to do so; generally either the Poisson is such that it would be at least as convenient to use the Poisson, or a normal approximation will suffice for both.

Glen_b
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  • How about the variance implication when approximating binomial by Poisson? – user3119750 Mar 20 '14 at 12:01
  • I'm not sure what you're after. When $\lambda=np$, you're approximating something with variance $np(1-p)$ by something with variance $np$, but $p$ is very close to zero, so that the percentage error in approximating $1-p$ by 1 is very small. – Glen_b Mar 20 '14 at 13:30