 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/discrete-mathematics-probability-and-continuous-mathematics/9/b/iv/solution

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

This is finite because

t - 1 > 1

. Here

ζ

is the Riemann zeta function on its convergent real domain. The floor function is nonnegative on the support, so the tail-sum identity applies directly.

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