What are the restrictions on the weights of a linear combination of a random sample as an unbiased estimator of E(X)

Question

I would like to find the proof of the following:

Let $X_1,...,X_n$ be a random sample of $X$ and $a_1,...,a_n$ real numbers.

If $\sum_{i=1}^n a_iX_i$ is an unbiased estimator of $E(X)$ then $\sum_{i=1}^n a_i = 1$

Answer 1

We need to know two things here:

1. The expectation is linear meaning that $\operatorname E[aX+bY]=a\operatorname EX+b\operatorname EY$.
2. If $X_1,\ldots,X_n$ is a random sample of $X$, then $\operatorname EX=\operatorname EX_1=\operatorname EX_2=\ldots=\operatorname EX_n$.

Using these two facts, we can show that $\sum_{i=1}^na_i$ has to be equal to $1$ if the estimator is unbiased.

I hope this is helpful and you can continue from here.