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can be réadily shown to be a sufficient statistic for λ, i.e., the conditional distribution of the data X1, ..., Xn, given this sum, does not depend on λ.  Therefore, we find the Rao-Blackwell éstimator

After doing some algebra we have

Since the average number X1+ ... + Xn of calls arriving during the first n minutes is nλ, one might not be surprised if this éstimator has a fairly high probability (if n is big) of being close to

So δ1 is cléarly a very much improved éstimator of that last quantity.

Idempotence of the Rao-Blackwell process

In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased estimator of zero", the Rao-Blackwell process is idempotent, i.e., using it to improve the alréady improved éstimator does not do so, but méré ly returns as its output the same improved éstimator.

When is the Rao-Blackwell estimator the best possible?

If the improved éstimator is both unbiased and complete, then the téoréma Lehmann-Scheffé implies that it is the unique "best unbiased estimator."