 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/6/a/iii/solution

The products

T U

and

U V

U

, so they are not independent. Using the definition of covariance and the independence of

T, U, V

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

By contrast,

T U

and

VW

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

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