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It is very apparent to me how using the normal distribution to estimate the probability of large, Poisson-distributed events may lead to significant underestimates of the probability of these events, especially when the mean event rate is low. This matches my intuition of a distribution with a heavy-tail. However, I fail to see how Poisson distributions verify the definition of a heavy-tailed distribution. Are Poisson distributions heavy tailed? If not, how does one define the slow probability decay of Poisson distributions with low mean?

Thank you.

deppep
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  • For $X\sim Poisson(\lambda )$, $kurtosis(X)=3+\dfrac{1}{\lambda}$. If $\lambda=\dfrac{1}{TREE(3)}$, then $kurtosis(X)$ looks pretty big to me! – Dave Jul 15 '22 at 10:44
  • This may be interesting https://stats.stackexchange.com/questions/86429/which-has-the-heavier-tail-lognormal-or-gamma – Florian Hartig Jul 15 '22 at 13:31
  • The same thing happens with the Bernoulli distribution, which is bounded, but has high kurtosis for small $p$ or small $1-p$. In such cases, the occurrence of the event with small probability may be called an "outlier." High kurtosis is a reflection of the outlier character: smaller $p$ implies that the rare event is more of an outlier. – BigBendRegion Aug 11 '22 at 16:26

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