Summation Rules Mistake?

Question

I'm looking at a solution for a stats problem and I think there is a mistake. The solution is as follows:

$$m_i^{-1} \sum_{e=1}^{m_i} u_{i,e} = m_i^{-1}\sum_{e=1}^{m_i}(f_i + u_{i,e}) = f_i + m_{i}^{-1}\sum_{e=1}^{m_i}v_{i,e}$$

I believe there was a mistake where they wrote $u$ instead of $v$ but aside from that I don't believe the last part makes sense. Am I misunderstanding or is this a mistake?

They show that $u_{i,e} = f_i + v_{i,e}$

Edit: I am of the opinion that $f$ should can't be added to the second term. I believe it needs to still multiply by everything and the value $m_i$.

Answer 1

It is actually right like this $-$ beside the $u$-$v$-typo of course. A sum where every element is independent of the index is similiar to adding up ones $($in case of definite sums$)$. I.e. we got that

$$\sum_{e=1}^{m_i}1=\underbrace{1+1+\dots+1}_{m_i~\text{times}}=m_i$$

Since we can simply pull out constants in such sums we may conclude that

$$m_i^{-1}\sum_{e=1}^{m_i}f_i=m_i^{-1}f_i\sum_{e=1}^{m_i}1=f_im_i^{-1}(m_i)=f_i$$

$$\therefore~m_i^{-1}\sum_{e=1}^{m_i}(f_i + v_{i,e})=m_i^{-1}\sum_{e=1}^{m_i}f_i+m_i^{-1}\sum_{e=1}^{m_i}v_{i,e} = f_i + m_{i}^{-1}\sum_{e=1}^{m_i}v_{i,e}$$