 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/6/b/iv/solution

For a nonnegative integer-valued random variable

W

E [W] = \sum_{k = 1}^{\infty} P (W \geq k)

W = \sum_{k \geq 1} 1_{{W \geq k}}

and using nonnegativity to interchange sum and expectation.

With

W = ⌊ X ⌋

, this gives

E [⌊ X ⌋] = k = 1 \sum \infty k^{1 - t} = ζ (t - 1) .

The Riemann zeta function is finite here because

t - 1 > 1

 New to topics? Read the docs here!