Solution Found!
Let X1, . . . , Xn be an i.i.d. sample from a Poisson
Chapter , Problem 68(choose chapter or problem)
Let \(X_{1}, \ldots, X_{n}\) be an i.i.d. sample from a Poisson distribution with mean \(\lambda\), and let \(T=\sum_{i=1}^{n} X_{i}\).
a. Show that the distribution of \(X_{1}, \ldots, X_{n}\) given T is independent of \(\lambda\), and conclude that T is sufficient for \(\lambda\).
b. Show that \(X_1\) 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.
Questions & Answers
QUESTION:
Let \(X_{1}, \ldots, X_{n}\) be an i.i.d. sample from a Poisson distribution with mean \(\lambda\), and let \(T=\sum_{i=1}^{n} X_{i}\).
a. Show that the distribution of \(X_{1}, \ldots, X_{n}\) given T is independent of \(\lambda\), and conclude that T is sufficient for \(\lambda\).
b. Show that \(X_1\) 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.
ANSWER:Step 1 of 3
Let 
We have to write,
Consider,
Where 
Therefore,
Where
It is does not dependent on the parameter 
