Interpreting formulas with $\Sigma \Sigma$ (two sigma)
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?
$\endgroup$ 62 Answers
$\begingroup$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).
$\endgroup$ 1 $\begingroup$To evaluate the double sum, first, expand the inner summation and then continue by computing the outer summation
$$\sum_{i=1}^4 \sum_{j=1}^3 ij = \sum_{i=1}^4(i + 2i +3i) = \sum_{i=1}^4 6i = 6 + 12 + 18 + 24 = 60$$
$\endgroup$
