Solved:
X
i
Distribution
does not mean what I thought

In brief: Subscripted random variables are not treated as iid draws from a given distribution.
​
What I expected:
n
∑
i=1
X
i
,{
X
i
NormalDistribution[μ,σ]}]
means “the sum of n iid N(μ,σ) random variables.” In this case, we obtain a variance of
2
nσ
.
​
Reality:
n
∑
i=1
X
i
,{
X
i
NormalDistribution[μ,σ]}]
is interpreted as
n
X
i
, i.e. “a constant times a single N(μ,σ).” In this case, we obtain a variance of
2
n
2
σ
.
​
​
If the interpretation were in line with my expectations, we would obtain a variance of
2
nσ
; in the latter case, we would obtain a variance of
2
n
2
σ
. We obtain the latter variance.
In[]:=
Simplify[Expectation[
2
(
n
∑
i=1
X
i
)
,{
X
i
NormalDistribution[μ,σ]}]-
2
Expectation[
n
∑
i=1
X
i
,{
X
i
NormalDistribution[μ,σ]}]
]
Out[]=
2
n
2
σ