Page 308 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 285

we can write

Thus, one has

which is free from p. In other words, is a sufficient statistic for the
unknown parameter p. !
Example 6.2.2 Suppose that X , ..., X are iid Poisson(λ) where λ is the
n
1
unknown parameter, 0 < λ < ∞. Here, χ = {0, 1, 2, ...}, θ = λ, and Θ = (0, ∞).
Let us consider the specific statistic . Its values are denoted by t
∈ = {0, 1, 2, ...}. We verify that T is sufficient for λ by showing that the
conditional distribution of (X , ..., X ) given T = t does not involve λ, what-
1
n
ever be t ∈ . From the Exercise 4.2.2 recall that T has the Poisson(nλ)
distribution. Now, we obviously have:

But, when , since is a subset of B = {T = t},
we can write

Hence, one gets