68. Let Xi, . . . , Xn be an 1.1.d. sample from a Poisson distribution with mean λ, and a. Show that the distribution of Xi, , Xn given T is independent of λ, and conclude that T is sufficient for λ....

Theorem A at Section 8.8.1 says
A necessary and sufficient condition for T(X1,..., Xn) to be sufficient for a parameter θ is that the joint probability function (density function or frequency function) factors in the form
f(x1,...,xn|θ) = g[T(x1,...,xn),θ]h(x1,...,xn)

Extracted text: 68. Let Xi, . . . , Xn be an 1.1.d. sample from a Poisson distribution with mean λ, and a. Show that the distribution of Xi, , Xn given T is independent of λ, and conclude that T is sufficient for λ. b. Show that X is not sufficient. c. Use Theorem A of Section 8.8.1 to show that T is sufficient. Identify the functions g and h of that theorem.