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Year 2 of the mathematics course of the University of Oxford

Options II course of the University of Oxford

Past exam of the options II course of the University of Oxford

2012

The same <beta integral> with one additional power of $y$ gives
$$
\mathbb E Y_{(r)}^2=\frac{r(r+1)}{(n+1)(n+2)}.
$$
Subtracting the square of the mean yields
$$
\operatorname{Var}(Y_{(r)})=\frac{r(n+1-r)}{(n+1)^2(n+2)}.
$$
At $r=1$ and $r=n$, therefore,
$$
\boxed{\operatorname{Var}(Y_{(1)})=\operatorname{Var}(Y_{(n)})=\frac n{(n+1)^2(n+2)}.}
$$
Their equality also follows by reflecting every uniform observation via $y\mapsto1-y$.

Solution

Past exam of the options II course of the University of Oxford / 2012 / O / O4 / b / ii / Solution

Past exam of the options II course of the University of Oxford / 2012 / O / O4 / b / ii / Solution

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