 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2012/o/o4/b/i/solution

For nonnegative integers

j, k

\int_{0}^{1} y^{j} (1 - y)^{k} d y = j! k! / (j + k + 1)!

. Multiplying the stated density by

y

and using this formula yields

E Y_{(r)} = \frac{n !}{( r - 1 )! ( n - r )!} \frac{r ! ( n - r )!}{( n + 1 )!} .

Cancellation gives

E Y_{(r)} = \frac{r}{n + 1} .

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