TITLE: Variance of a sum of IID random variables.
QUESTION [5 upvotes]: Let's say we have a sequence of $n$ IID random variables, $I_i$. Let's define a new random variable which is their sum:
$$S = \sum_i I_i$$
To calculate the variance of $S$ we can say - 
$$V(S) = \sum_i V(I_i) = nV(I_1)$$
Or we can also say that $S$ has the same distribution as $nI_1$
So, we should have $V(S) = V(nI_1) = n^2V(I_1)$.
This is of course, in direct contradiction to what we got above. What am I missing?

REPLY [9 votes]: You're missing the difference between these two things:
$$
\begin{align}
& I_1 + I_2+ \cdots + I_n, \\[8pt] & I_1 + I_1+ \cdots + I_1.
\end{align}
$$
In one of those, the terms are independent; in the other, they're as far from independent as they can get.
NOTE: These do not both have the same distribution.