 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2008-printed-variant/o/o5/ii/solution

For each normal sample, its sample mean is independent of its sample variance, and the scaled residual sum of squares is chi-squared. Independence of the samples also makes their two residual sums independent. Adding their chi-squared distributions gives

\frac{( m - 1 ) S _{X}^{2} + ( n - 1 ) S _{Y}^{2}}{σ ^{2}} \sim χ_{m + n - 2}^{2} .

Consequently

d = (m + n - 2) / σ^{2}, d S^{2} \sim χ_{m + n - 2}^{2} .

The pooled variance is also independent of the difference of sample means.

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