 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2010/o/o4/b/ii/solution

A linear combination of independent normal variables is normal; the corresponding asymptotic statement follows from the joint central limit theorem and the continuous mapping theorem. Thus

p - q \approx N (p - q, \frac{p ( 1 - p )}{2 M} + \frac{q ( 1 - q )}{2 N}) .

The two variances add because of independence. This approximation assumes both sample sizes grow and the true probabilities are interior.

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