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.

 

[blue_raver22]

Hello there,

Calculating the sum of correlated random variables can be a bit tricky, but here are the steps you can follow:

1. First, determine the correlation between each pair of random variables. This can be done by finding the covariance between each pair and dividing it by the product of their standard deviations.

2. Once you have the correlation between each pair, you can use the formula for the sum of correlated random variables:

T = Ʃ Xi + Ʃ Ʃ ρij * Xi * Xj

where ρij is the correlation between Xi and Xj.

3. You can then simplify the formula by factoring out the common random variables and only keeping the ones that are unique.

T = Ʃ Xi + Ʃ Ʃ ρij * Xi * Xj

= Ʃ Xi + Ʃ Xi * Ʃ ρij * Xj

4. Finally, you can calculate the sum by substituting in the values of the random variables and their correlations.

I hope this helps! Let me know if you have any further questions.