A summation formula where I have trouble to find the priorities of its calculations

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

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" ?
• 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. Commented Apr 12, 2022 at 9:43
• 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 $(x/[y)/z]$. Commented Apr 12, 2022 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. Commented Apr 12, 2022 at 21:32

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.