# a problem on normal approximation to the poisson - from Durrett's book

I came across a problem from Durrett's book Elementary Probability

P.217 Example 6.26

Normal approximation to the Poisson. Each year inMythica, an average of 25 letter carriers are bitten by dogs. In the past year, 33 incidents were reported. Is this number exceptionally high?

Solution: Assuming that dog bites are a rare event, we use the Poisson distribution for the number of dog bites. As we observed in Example 6.3, a Poisson with mean 25 is the sum of 25 independent Poisson mean 1 random variables, so we can use the normal to approximate the Poisson.

The mean is $25$, while the standard deviation is $\sqrt{25} = 5$.Writing the observed event as $≥32.5$ we see that this is $(32.5−25)/5 = 1.5$ standard deviations above the mean, so the normal approximation is $$P(\chi ≥ 1.25) = 1 − 0.8943 = 0.1057$$so this is not a very unusual event.

Why it is $P(\chi ≥ 1.25)$, not $P(\chi ≥ 1.5)$? Can anybody explain it to me?

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## 1 Answer

Because even very good books can have typos. The number of typos in a page is often modeled using the Poisson distribution.

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which, under some mild condition, you can approximate with a normal distribution! – Jean-Sébastien Nov 18 '12 at 5:47
But $P(\chi ≥ 1.5) = 1- 0.9332 = 0.0668$, so it's less likely to be just typos – xoofee Nov 18 '12 at 5:47
"Typo" is sometimes used as a polite word for "minor slip," which is itself sometimes a polite term for "mistake." – André Nicolas Nov 18 '12 at 5:50
@AndréNicolas: Aha, I see – xoofee Nov 18 '12 at 6:02