Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 18H / a / Solution

For independent $U_1,U_2\sim\operatorname{Uniform}(0,1)$, the sum has triangular density
$$
f(s)=
\begin{cases}
s,&0\leq s\leq1,\\
2-s,&1\leq s\leq2,\\
0,&\text{otherwise}.
\end{cases}
$$
Its upper tail for $1\leq c\leq2$ is $(2-c)^2/2$.


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 Incoming links: Sum of two independent uniform variables