Calculate T: Sum of Correlated Random Variables from i=1 to m

[starblazzers]

Hi all, I would like to get assistance on how to obtain the sum of correlated random variables

T = Ʃ Xi, from i=1 to m

where Xi are correlated rvs


Please help if you can!

 

[Stephen Tashi]

how to obtain the sum of correlated random variables

That doesn't make sense as a question. If you want the sum, you just take the sum.

Perhaps you are trying to ask something about the mean of the sum or the variance of the sum.

 

[Stephen Tashi]

The expectation (i.e. mean) of a sum of random variables is equal to the sum of their means. It doesn't matter whether the random variables are correlated or not.

The variance of a sum of random variables is the sum of all the pairwise covariances, including each variable paired with itself (in which case, the variance of that variable is computed).

Let $X_1, X_2,...X_n$ be random variables.
Let $S = \sum_{i=1}^n X_i$
Let the expectation of a random variable $X$ be denoted by $E(X)$
Let the variance of a random variable $X$ be denoted by $Var(X)$
Let the covariance of a random variable $X$ be denoted by $Cov(X)$
(So $Var(X) = Cov(X,X)$ . )

Then
$E(S) = \sum_{i=1}^n E(X_i)$

$Var(S) = \sum_{i=1}^n ( \sum_{j=1}^n Cov(X_i,X_j) )$

 

[Stephen Tashi]

How would the mean E(S) and Var(S) be if n is also a random variable?

I don't know any simple formula that applies. There could be simple formulas in special cases. For example if the means of the $X_i$ are all the same and $n$ is independent of each of the $X_i$ then I think the mean of $S$ is given by the product: (the mean of $n$ ) (the mean of $X_1$ ).


As an example of a case where $n$ is dependent on the $X_i$, suppose the sum is formed according to the rule: Set the sum = $X_1$ and then add another $X_i$ until you draw some $X_i > 2.0$. When that happens, stop summing.