Assuming you are summing 10 poisson variables, and the sum of their results is 60, how would you approximate a 95% CI for theta?

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Evandassy8yp · Accepted Answer

Step 1The sample mean would give you a point estimate for the rate parameter. What you need in order to calculate a confidence interval is a variance of the sampling distribution.Formally, if&amp;nbsp;x=(x1,…,xn)&amp;nbsp;is an IID sample drawn from a Poisson distributionX~Poisson(θ),&amp;nbsp;&amp;nbsp;Pr[X=x]=e-θθxx! &amp;nbsp;&amp;nbsp;x=0,1,2,…,and we wish to find a 95% CI for&amp;nbsp;θ, we can choose the point estimate based on the method of maximum likelihood:θ^MLE=x―,which is a result I leave as an exercise. Then we would calculate the variance of this estimator:Var[θ^MLE]=Var1n∑i=1nXi=iidθn,hence the estimated variance of the estimator isVar^[θ^MLE]=x¯n,and an asymptotic 2-sided&amp;nbsp;100(1−α)%&amp;nbsp;Wald-type confidence interval for&amp;nbsp;θ&amp;nbsp;would then bex¯±zα/2*x¯n,where&amp;nbsp;z{α2}⋅&amp;nbsp;is the upper&amp;nbsp;α2&amp;nbsp;quantile of the standard normal distribution. Of course, this is not the only confidence interval one can construct; this one, as pointed out above, is approximate and based on the asymptotic distribution of&amp;nbsp;θ^.

Assuming you are summing 10 poisson variables, and

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