# Sufficiency for a binomial population

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?

b) The question is unclear in that it mixes sufficiency and estimation: once you have produced a sufficient statistic, it is an estimator of anything you chose. To make it precisely an estimator of $\theta$, this sufficient statistic must enjoy some additional property like, e.g., unbiasedness. Now it may just be that the question is to determine whether or not$$\frac{X_1+2X_2}{n_1+2n_2}$$ is sufficient. A further hint is that, since$$\frac{X_1+X_2}{n_1+n_2}$$is sufficient, both sufficient statistics should be in one-to-one correspondence.