Show that $\frac{1}{n-1}\sum_{i=1}^n(X_iY_i-\overline{X}\overline{Y})$ is an unbiased estimator of $\text{Cov}[X,Y].$

Question

Assume that $(X_1,Y_1),...,(X_n,Y_n)$ is a sample on a two-dimensional random variable $(X,Y)$ and that $E[X^2], \ E[Y^2]$ and $E[XY]$ are all finite so that the variances and covariance are well-defined. Show that

$$S=\frac{1}{n-1}\sum_{i=1}^n(X_iY_i-\overline{X}\overline{Y})$$ is an unbiased estimator of $\text{Cov}[X,Y].$

So I'm supposed to show that $E[S]=\text{Cov[X,Y]}.$ Now

$$E\left[\sum_{i=1}^n(X_iY_i-\overline{X}\overline{Y})\right]=\sum_{i=1}^nE[X_iY_i]-nE[\overline{X}\overline{Y}]=n(E[XY]-E[\overline{X}\overline{Y}]).\tag1$$

According to the formula for variance the last step in $(1)$ can be rewritten as

$$n(\mu_x\mu_y+\text{Cov}[X,Y]-(\mu_x\mu_y+\text{Cov}[\overline{X}\overline{Y}]))=\text{Cov}[X,Y]-\text{Cov}[\overline{X}\overline{Y}])), \tag2$$

where $\mu_x$ and $\mu_y$ are the respective mean.

Questions:

1. In $(2)$, I understand that from the covariance formula we get $$\mu_x\mu_y=E[X]E[Y]=E[XY]-\text{Cov}[X,Y],$$ but why is it also equal to $$\mu_x\mu_y=E[X]E[Y]=E[XY]-\text{Cov}[\overline{X}\overline{Y}]?$$ Shouldn't the sample means with overline cause any difference?

2. How do I proceeed from here?

Answer 1

For the first part, see that $E(\overline X)=\mu_x$ and $E(\overline Y)=\mu_y$ because: $$ E(\overline X)=E(\frac 1n\sum_i X_i)=\frac 1n\sum E(X_i)=\frac 1n (n\mu_x)=\mu_x. $$ For the second question, the solutions follows as: $$ \text{Cov}[\overline X\overline Y]=E[(\overline X-\mu_x)(\overline Y-\mu y)]=E(\overline X\overline Y)-\mu_x\mu_y. $$ Now see that $$ E(\overline X\overline Y)=\frac{1}{n^2}\sum_{i,j}E(X_iY_j)=\frac 1n E(XY)+\frac{n-1}{n}\mu_x\mu_y. $$ Hence: $$ \text{Cov}[\overline X\overline Y]=\frac 1n E(XY)-\frac 1n \mu_x\mu_y=\frac 1n \text{Cov}(XY). $$ Plug this in and you have: $$ E(S)=\frac{1}{n-1}\times n\times (\text{Cov}(XY)-\frac 1n \text{Cov}(XY))=\text{Cov}(XY). $$