Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 9 / a / iii / Solution

For the overlapping products, factor using the independence of

T, U, V

:

Cov (T U, U V) = E [T U^{2} V] - E [T U] E [U V] = 1 \cdot 2 \cdot 1 - 1 \cdot 1 = 1.

This is the covariance of overlapping independent products: the shared

U

contributes its second moment. In contrast,

T U

and

VW

use disjoint independent factors and are themselves independent, giving

Cov (T U, U V) = 1, Cov (T U, VW) = 0.

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