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Could you please tell me what I am supposed to do when facing two sigma ($\Sigma \Sigma$), such as in the covariance formula?

Should I sum the results obtained by each or rather multiply them?

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    Could you give an example so that your question becomes clearer?2012-06-04
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    As stated above an example is the formula for the variance. It looks like this ∑∑... where the first sigma goes from i=1 to n and the second from j=1 to r . Thanks @m-turgeon for answering!2012-06-04
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    What are you summing? In some particular cases, we can simplify.2012-06-04
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    Salut @davide , ce n'est pas un cas particulier qui m'intéresse ici. en effet je voulais juste savoir comment faut-il se comporter face à un double sigma. dois-je sommer ou bien multiplier les résultats obtenus ? O preferivi in italiano? Grazie in anticipo.2012-06-04
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    Non importa (anche se qui la linguia ufficiale è l'inglese, dunque continuo in inglese). If you can write $a_{ij}=b_i\cdot c_j$, then you just multiply the sum over $i$ with the sum over $j$. But is doesn't need to be the case in general. A good way to think about double summations is summation over an array. You can first sum with respect to the rows, then with respect to the columns (and also switch the two sums).2012-06-04
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    thanks indeed @DavideGiraudo2012-06-04

2 Answers 2

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This notation has the following meaning: consider real numbers $a_{ij}$, where $i$ ranges from 1 to 3, and $j$, from 1 to 2. We thus have the following equality:

$$\sum_{i=1}^3 \sum_{j=1}^2 a_{ij}=\sum_{i=1}^3(a_{i1}+a_{i2})=(a_{11}+a_{12})+(a_{21}+a_{22})+(a_{31}+a_{32}).$$

There is nothing more to it. However, as Davide pointed out, sometimes you can simplify the computations - but this highly depends on the nature of the number $a_{ij}$ (for example, they might satisfy a recurrence relation).

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    But, imo, it's not always easy to make an idea of the meaning of the formulas involving double (or more) "sigmas", especially when the ranges vary quite a lot or are connected somehow. I think that this confusion may come from a not enough experience in reading these formulas. Apart from this, do you have any suggestions on how to "immediately" grasp the meaning of the formulas? Do you usually "immediately" understand the formulas with these "sigmas" or you also take some time to interpret them? So that I can make an idea of my level of interpretation in this context.2016-12-19
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    @nbro I sometimes write out a few terms, to give me an idea of what the sum looks like. Do you have an example in mind that was particularly difficult to parse?2016-12-19
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    Not at the moment, but I would cheerfully read an article talking about the topic, i.e. explaining using examples how to overcome or try to overcome the difficulties in interpreting this notations.2016-12-19
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To evaluate the double sum, first, expand the inner summation and then continue by computing the outer summation

$$\sum_{i=1}^4 \sum_{j=1}^3 = \sum_{i=1}^4(i + 2i +3i) = \sum_{i=1}^4 6i = 6 + 12 + 18 + 24 = 60$$