Modifying a point-estimation to get a $unbiased$ $estimator$ $of$ $variance$

Question

My Exercise Problem:

Let x1, x2, ..., x7 be observations of independent random variables X1, X2, ..., X7 with E(Xi) = μ and Var(Xi) = σ2, for all i = 1, 2, ..., 7.
Let σ²_obs = $c(x_2^2 + x_6^2- \frac{(x_1 + x_7)^2}{2})$ be a point estimation of σ².
Determine the constant c such that σ²_obs becomes a unbiased estimator of variance.

The correct solution:
c = 1

What I need help with
How do I solve this problem? I am very new to point estimations and "unbiased estimators of variance and standard deviations". I want to gradually become better at solving these type of questions, where should I start? Perhaps it would be good for me to start with similar problems but on a easier level, any advice where I can find such problems?

Answer 1

Considering that $E(X^2)=V(X)+E^2(X)$ and setting

$$T=X_2^2+X_6^2-\frac{(X_1+X_7)^2}{2}$$

You get that

$$E(T)=2(\sigma^2+\mu^2)-\frac{1}{2}(2\sigma^2+4\mu^2)=\dots=\sigma^2$$

Thus T (your original estimator with $c=1$) is unbiased of variance, as requested