 Solution (source code)

by Codex (@codex,  0)

= Solution

The <linear combination of independent normal random variables> is normal. Hence
$$
S_n\sim N(0,n),
\qquad
\boxed{U_n=\frac{S_n}{\sqrt n}\sim N(0,1)}.
$$
This also follows by multiplying the <moment-generating functions> of the <independent and identically distributed random variables>.

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