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I'm unease with summations and their priorities.
I'm stumbling upon this, on a book:

enter image description here

Am I :

  1. correctly reading this, here :

$$ \text{MSI} = \sum\limits_{i}\Bigg[\frac{\Bigg(\frac{p_{i,j}}{\sqrt{\prod\limits_{i}{a_{i,j}}}}\Bigg)}{n_{i}}\Bigg] $$

  1. believe rightly that $\prod\limits_{i}{a_{i,j}}$
    is a "secondary" (underlying?) complete loop that I have to perform on all the $a$ items to multiply them first, whatever the same $i$ index is used elsewhere in this formula?
    or, at the opposite, does it mean: "don't make this $i$ under the product go over the $i$ value that you are currently considering on the summation" ?
Marc Le Bihan
  • 171
  • It is clearly bad practice to use the same index $i$ in $\sum$ and $\prod$. Most likely it is a typo. It is also bad practice not to mention where the summation and product indices start and end. – Kurt G. Apr 12 '22 at 09:43
  • Your notation is confusing. Are the square brackets intended to include the $p_{i,j}$? If so, they need to be taller. If not, then the fraction line below $p_{i,j}$ needs to be wider. As it is, the grouping is ambiguous. It's as if you wrote $(x/[y)/z]$. – Jair Taylor Apr 12 '22 at 20:12
  • @JairTaylor I am at the maximum of the brackets size, with \Bigg[. I don't believe that taller is possible with latex. – Marc Le Bihan Apr 12 '22 at 21:32

1 Answers1

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This formula is the same as \begin{align*} \mathrm{MSI}=\sum_{i}\frac{p_{i,j}}{n_i\sqrt{\prod_{\color{blue}{k}}a_{\color{blue}{k},j}}} \end{align*}

Here we replaced the index variable $i$ which belongs to the scope of the product symbol with $k$.

  • Note $k$ is a bound variable with scope given by the product operator.

  • The index variable $i$ is a bound variable with scope given by the sum.

  • The index variable $j$ is a free variable, neither bound by the summation symbol nor by the product symbol.

As indicated by @KurtG it is considered a bad style, seriously violating readability to use the same index variable at the same place with different meaning.

Markus Scheuer
  • 108,315