I am using a book, where the following exercise appears:
Two estimators for a random sample of size $n$ (Bernoulli population):
$T_1=\frac{\sum\limits_{i=1}^n X_i + 2X_n}{n+2}$
$T_2=\frac{\sum\limits_{i=1}^{n-2} X_i + 2X_n}{n+2}$
I want to compare them in terms of their relative efficiency.
I know that:
$VAR[T_2]=VAR\bigg[\frac{\sum\limits_{i=1}^{n-2} X_i + 2X_{n}}{n+2}\bigg] \\ =\frac{1}{(n+2)^2}.\{VAR[X_1+X_2+...+X_{n-2}]+VAR[2X_n]\} \\ $
however I am not sure how to continue... the variance of a Bernoulli is $p.q$, should I replace each $X_i$ with $p.q$?
The book suggests:
(1.1) $= \frac{1}{(n+2)^2}.[(n+2).pq+4pq] \\ = \frac{1}{(n+2)^2}.(n+2).pq \\ = \frac{pq}{n+2}$
however, I do not understand these three steps (1.1).
The same for the estimator $T_1$:
$VAR[T_1]=VAR\bigg[\frac{\sum\limits_{i=1}^{n} X_i + 2X_n}{n+2}\bigg] \\ =\frac{1}{(n+2)^2}.\{VAR[X_1+X_2+...+X_n+2X_n]\}$
the book suggests:
(2.1) $=\frac{1}{(n+2)^2}.\{VAR[X_1+X_2+...+X_{n-1}+3X_n]\} \\ = \frac{n+8}{(n+2)^2}.pq$
I do not understand these steps (2.1).
Sorry for the very basic question.