I am "new" to probability/statistics and I was hoping someone could verify that this is correct. Let $Y_1,\ldots,Y_n$ be random variables that follow a common distribution with mean $\mu$ and variance $\sigma^2$. However, I am not assuming that the variables are independent. I want to compute the expected value and variance of $X=Y_1+\cdots+Y_n$. Then,
$$\mathbb{E}(X) = \mathbb{E}\left(\sum_{i=1}^{n} Y_i \right) = \sum_{i=1}^{n} \mathbb{E}\left( Y_i \right) = n \cdot \mu.$$
Moreover,
\begin{eqnarray*}
\operatorname{Var}(X) &=& \operatorname{Var}\left(\sum_{i=1}^{n} Y_i \right) \\
&=& \sum_{i=1}^{n} \operatorname{Var}\left( Y_i \right) + 2\cdot\left(\sum_{1\leq i<j\leq n} \operatorname{Cov}(Y_i,Y_j) \right) \\
&=& \sum_{i=1}^{n} \operatorname{Var}\left( Y_i \right) + 2\cdot\left(\frac{ n(n -1 )}{2}\right) \cdot C \\
&=& n \cdot \sigma^2 + n(n -1)\cdot C\\
&=& n\cdot ( \sigma^2 + (n-1)\cdot C),
\end{eqnarray*}
where we have used the properties of the variance, and the fact that for any $i\neq j$, the variables $Y_i,Y_j$ follow the same distribution, and so $\operatorname{Cov}(Y_i,Y_j)=C$ for all $i\neq j$, for some constant $C$.
Question 1: Is this right? I have an uneasy feeling about assuming that all the covariances equal a constant C. Is there some additional hypothesis on the original random variables $Y_1,\ldots,Y_n$ that one needs to assume so that the covariance $\operatorname{Cov}(Y_i,Y_j)=C$ independent of the chosen $i\neq j$?
Question 2: Also, is $C=\operatorname{Cov}(Y_i,Y_j)=\operatorname{Cov}(Y_i,Y_i)=\sigma^2$?
Thanks!
Best Answer
Let $n = 3$. $Cov(Y_1,Y_2), Cov((Y_1,Y_3), Cov(Y_2,Y_3)$ can all be the same or different. They can take any values, so long as the 3 by 3 covariance matrix is positive semi-definite.
A very common case, much studied on this board, would be a 3 dimensional Normal, which in your case of identically distributed components would have equal elements on the diagonal. The covariances between the components (and in this case, therefore also the correlations between the components) can be the same or different, as long as the covariance matrix is positive semi-definite.
As @whuber has hinted in a comment, while the scenario in your question 2 would be possible, and corresponds to perfectly correlated variables, which corresponds to a covariance matrix in which all entries are equal (and therefore the covariance matrix is singular), it certainly need not be the case.