# Solved – Confusion about confidence interval around mean of poisson distribution

This question is based on this question: How to calculate a confidence level for a Poisson distribution? and its answers.

In that post, the question is "How do I calculate the confidence interval of a poisson distribution with \$n = 88\$ and \$lambda =47.18\$?"

\$\$ lambda pm 1.96sqrt{dfrac{lambda}{n}}, \$\$

for the upper and lower bounds. It might be noted that this is an approximation which is okay when \$nlambda\$ is big enough — whatever big enough might be, apparently \$4152\$ is big enough.

Now, as far as I know — this might be completely wrong since I haven't really properly studied statistics yet — the confidence interval gives you an interval such that the probability that the mean is in this interval is \$95%\$. So, I'd think the probability of being outside this interval would be \$2.5%\$? But it's not. So I'm confused.

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• "the confidence interval gives you an interval such that the probability that the mean is in this interval is 95%"

This is not quite true. Indeed, in this framework, the unknown mean is fixed (either it lies inside the interval or it lies outside, there is no probability attached to it). Instead, you should say that the probability that the interval (which is random) covers the true mean equals 95%. Or that there is a 5% chance that the interval does not cover the true mean.

• "Ok, but the confidence interval with using the approximate technique above gives (45.7467, 48.617). Yet for λ=47.18 I get P(X≥49)=0.41469. That I don't get." (cf. comment below question)

\$(45.7467, 48.617)\$ is meant to provide information concerning the mean of the Poisson distribution, i.e. \$lambda\$. Not about an observation from the Poisson distribution.

Perhaps you are interested in a prediction interval