<s>[INST] Confusion with Summation notation I need to compute the value of this:
$$\frac{1}{3}\left(\sum_{n=1}^{3} \frac{z_{n}}{f} + N(2,2)\right)$$
the $N$ is a gaussian noise with mean=2 and standard deviation=2. 
The question:
is this equivalent to:
$$
\frac{1}{3}\left(\frac{z_{1}}{f} + N(2,2)+\frac{z_{2}}{f} + N(2,2)+\frac{z_{3}}{f} + N(2,2)\right)
$$
or 
$$
\frac{1}{3}\left(\left(\frac{z_{1}}{f} +\frac{z_{2}}{f} +\frac{z_{3}}{f}\right) + N(2,2)\right)
$$
 [/INST] 
The scope of the sigma operator $\Sigma$ is solely defined via arithmetic precedence rules. The scope is given by the expression that follows immediately the $\Sigma$ and is valid respecting the arithmetic precedence rules up to an operator with precedence level equal to '$+$' or up to the end if no such operator follows.
This implies that
\begin{align*}
\color{blue}{\sum_{n=1}^{3} \frac{z_{n}}{f} + N(2,2)}&=\sum_{n=1}^{3}\left( \frac{z_{n}}{f} \right)+ N(2,2)\\
&=\left(\sum_{n=1}^{3} \frac{z_{n}}{f}\right) + N(2,2)\\
&\,\,\color{blue}{=\frac{z_{1}}{f} + \frac{z_{2}}{f} + \frac{z_{3}}{f} + N(2,2)}\\
\end{align*}

Hint: You might find chapter 2: Sums in Concrete Mathematics by R.L. Graham, D.E. Knuth and O. Patashnik helpful. It provides a thorough introduction in the usage of sums.
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