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Year 1 of the computer science course of the University of Oxford

Past exam of the year 1 of the computer science course of the University of Oxford

2019

Discrete mathematics, continuous mathematics, and probability

Split the unit square into $v\le u$ and $v>u$. The maximum is $u$ on the first triangle and $v$ on the second, so
$$
\begin{aligned}
\mathbb E[U\max\{U,V\}]
&=\int_0^1\left[\int_0^u u^2\,dv+\int_u^1 uv\,dv\right]du\\
&=\int_0^1\left[u^3+\frac u2(1-u^2)\right]du\\
&=\frac18+\frac14=\boxed{\frac38}.
\end{aligned}
$$
The shared boundary has zero <probability>, so it is not counted twice in this <joint density> calculation.

Solution

Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 7 / b / ii / Solution

Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 7 / b / ii / Solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 7 / b / ii / Solution

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