The question at hand is: If X1 and X2 are independent random variables having binomial distributions with the parameters θ and n1 and θ and n2,
a ) show that (X1 +X2/n1+n2) is a sufficient estimator of θ. The textbook I am using shows how to find sufficient statistics as it relates to random variables from a Bernoulli population. Since the Bernoulli and binomial distributions are related I was wondering if I could use the Bernoulli to solve this? This is what the textbook does in their example:
(This image is from John E Freund's Mathematical Statistics with Applications 8th edition, Chapter 10 page 295)
Would I take the same approach?
b) In reference to part a, is (X1 +2X2)/(n1 +2n2) a sufficient estimator of θ? I am not understanding how to do this. Would I follow the same procedure as part a?