Covariance of variable and a mean of variables

Question

Here is the covariance of a random variable with the mean of $n$ random variables (of which it is one):

$cov (a_{i},\bar{a}_n)$

(where $a_{i}$ is a random variable, and there are $n$ of these variables)

I am wondering whether this can be (a) rewritten conveniently in matrix alegra format and (b) treated by a general solution?

Many thanks if you can help enlighten me!

Answer 1

Did you intend $\bar a_n$ to be the sample mean? $\displaystyle~~\bar a_n = \sum\limits_{j=1}^n \dfrac{a_j}{n-1}$?

Then: $$\begin{align}\mathsf {Cov}(a_i, \bar a_n) = & ~ \mathsf E(a_i\bar a_n)-\mathsf E(a_i)\mathsf E(\bar a_n) \\[2ex] = & ~ \mathsf E\left(a_i\sum\limits_{j=1}^n \dfrac{a_j}{n-1}\right)-\mathsf E(a_i)\mathsf E\left(\sum\limits_{j=1}^n\dfrac{ a_j}{n-1}\right) \end{align}$$

Simplify and complete.

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Hint:

Use the Linearity of Expectation and also that: $\mathsf E(a_ia_j)=\begin{cases}\mathsf E(a_i^2) & : i=j \\ \mathsf E(a_i)~\mathsf E(a_j) & : i\neq j\end{cases}$