Sufficient statistics for bernoulli distribution

Question

Let $Y_1, \ldots, Y_n $ be a random sample of size $n$ where each $Y_i \sim \textrm{Bernoulli}(p), $ and let $Y = \sum Y_i $ for $i = 1, \ldots, n.$

The estimator is $W= (Y+1)/(n+2). $

Is the estimator a sufficient statistics for parameter p?

I wanted to use the factorization theorem for this problem and wrote out the joint pdf, but I was stuck on rearranging the joint pdf

$$f(y_1,\ldots, y_n;p)= \prod p^{y_i}(1-p)^{1-y_i}. $$

Answer 1

$$f(y_1...y_n;p)= \prod^n_{i=1} p^{y_i}(1-p)^{1-y_i}=$$

$$ = p^{y_1+...+y_n}(1-p)^{n-(y_1+...+y_n)}= $$

$$ =p^{\sum y_i}(1-p)^{n-\sum y_i} $$

By Neyman-Fisher factorization theorem the statistic $Y$ is sufficient for parameter $p$.

$Y = (n+2)W-1$, i.e. $W$ is a one-to-one function of a sufficient statistic $Y$. Hence, $W$ is also a sufficient statistic.