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>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 1 / 12F / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ia/paper-1/12f/b/solution