Assume we that we calculate the expected value of some measurements $x=\dfrac {x_1 + x_2 + x_3 + x_4} 4$. what if we dont include $x_3$ and $x_4$, but instead we use $x_2$ as $x_3$ and $x_4$. Then We get the following expression $v=\dfrac {x_1 + x_2 + x_2 + x_2} 4$.
How do I know if $v$ is a unbiased estimation of $x$?
I am not sure how to approach this problem, any ideas are appreciated!
On
Let $\theta$ be some parameter. Let $X$ be an estimator.
$X$ is called an unbiased estimator for $\theta$ if $E[X] = \theta$.
Note that $X$ is a random variable (or random vector) while $\theta$ would be a scalar (or vector).
Let's say $x_1$ and $x_2$ are random variables with $E[x_1] = E[x_2] = \mu$. Then estimator $y = \frac{1}{5} x_1 + \frac{4}{5} x_2$ would be an unbiased estimate of $\mu$ since $E[y] = \mu$.
[EDIT: Assumed $x_k$ are random variables.]
We say $v$ is an estimator of random variable $x$ if $E[v]=E[x]$, where $E[\cdot]$ is expectation of random variables.
Recall that expectation is a linear operator, i.e. if $X$ and $Y$ are random variables and $a,b$ are constants, $E[aX+bY]=aE[X]+bE[Y]$. So we get $E[x]=\frac{1}{4}(E[x_1]+E[x_2]+E[x_3]+E[x_4])$, and $E[v]=\frac{1}{4}(E[x_1]+3E[x_2])$. Notice that $E[x]=E[v]$ is equivalent to $E[x]-E[v]=0$.
So $v$ is an unbiased estimator if $E[x]-E[v]=0 \Leftrightarrow E[x_3]+E[x_4]-2E[x_2]=0.$