 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2011/m/3/c/ii/solution

The identity

Y = a X + W

b = 1 - 2 a

and

c = 1 - a

. Independence will follow from zero covariance, because

X, W

Cov (X, W) = 2 b + c = 3 - 5 a .

Thus

a = \frac{3}{5}, b = - \frac{1}{5}, c = \frac{2}{5} .

These values give

W = (- Z_{1} + 2 Z_{2}) /5

1/5

and zero covariance with

X

. Joint normality then proves the requested independence.

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