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.