T = Ʃ Xi, from i=1

  • #1
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!
 

Answers and Replies

  • #2
Stephen Tashi
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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.
 
  • #3
Stephen Tashi
Science Advisor
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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 [itex] X_1, X_2,...X_n [/itex] be random variables.
Let [itex] S = \sum_{i=1}^n X_i [/itex]
Let the expectation of a random variable [itex] X [/itex] be denoted by [itex] E(X) [/itex]
Let the variance of a random variable [itex] X [/itex] be denoted by [itex] Var(X) [/itex]
Let the covariance of a random variable [itex] X [/itex] be denoted by [itex] Cov(X) [/itex]
(So [itex] Var(X) = Cov(X,X) [/itex] . )

Then
[itex] E(S) = \sum_{i=1}^n E(X_i) [/itex]

[itex] Var(S) = \sum_{i=1}^n ( \sum_{j=1}^n Cov(X_i,X_j) ) [/itex]
 
  • #4
Stephen Tashi
Science Advisor
7,583
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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 [itex] X_i [/itex] are all the same and [itex] n [/itex] is independent of each of the [itex] X_i [/itex] then I think the mean of [itex] S [/itex] is given by the product: (the mean of [itex] n[/itex] ) (the mean of [itex]X_1[/itex] ).


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

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