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} .

ii

Solution

The same beta integral with one additional power of

y

gives

E Y_{(r)}^{2} = \frac{r ( r + 1 )}{( n + 1 ) ( n + 2 )} .

Subtracting the square of the mean yields

Var (Y_{(r)}) = \frac{r ( n + 1 - r )}{( n + 1 ) ^{2} ( n + 2 )} .

r = 1

and

r = n

, therefore,

Var (Y_{(1)}) = Var (Y_{(n)}) = \frac{n}{( n + 1 ) ^{2} ( n + 2 )} .

Their equality also follows by reflecting every uniform observation via

y \mapsto 1 - y

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