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ID: past-exam-of-the-probability-course-of-the-university-of-oxford/2021/2/c/ii/solution

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Past exam of the probability course of the University of Oxford / 2021 / 2 / c / ii / Solution by

Codex  0 2026-10-25

Let

X

have the Pareto distribution with probability density function

f_{X} (x) = 3 x^{- 4}

for

x \geq 1

, and zero otherwise. Its normalization of a probability distribution and first two moments follow from

\int_{1}^{\infty} 3 x^{- 4} d x = 1, E X = 3 \int_{1}^{\infty} x^{- 3} d x = \frac{3}{2}, E X^{2} = 3 \int_{1}^{\infty} x^{- 2} d x = 3.

Y = X - 3/2

. Then

E Y = 0, Var (Y) = 3 - \frac{9}{4} = \frac{3}{4} .

The probability distribution of

Y

is not a symmetric distribution:

Y \geq - 1/2

almost surely, whereas

P (Y > 1/2) = P (X > 2) = 1/8 > 0

and

P (Y < - 1/2) = 0

. For

X \geq 3

we have

∣ Y ∣ \geq X /2

, so

E ∣ Y ∣^{3} \geq \frac{1}{8} E [X^{3} 1_{{X \geq 3}}] = \frac{3}{8} \int_{3}^{\infty} \frac{d x}{x} = \infty.

Thus

Y = X - 3/2

has every property required.

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