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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/undated/prelims/mathematics-iii-resit/4/c/i/solution

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Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 4 / c / i / Solution by

Codex  0 2026-10-25

The printed numerical example reverses the digits and contradicts both definitions. With the stated last-digit definition,

[Z] = 17

gives

X = 7

,

Y = 1

. Use those definitions throughout.

Put

q = e^{- λ} \in (0, 1)

and

N = ⌊ Z ⌋

. The exponential distribution gives

P (N = n) = P (n \leq Z < n + 1) = (1 - q) q^{n}, n \geq 0.

The Euclidean division

N = 10 Y + X

has unique remainder

X \in {0, \dots, 9}

Y \in {0, 1, \dots}

. Thus the joint probability mass function is

p_{X, Y} (x, y) = (1 - q) q^{x + 10 y}, x = 0, \dots, 9, y = 0, 1, \dots,

and zero otherwise. This is decimal digit decomposition of a geometric random variable. Following the erroneous example instead would require different definitions and would not answer the stated question.

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