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Variance of an audit sample

Expected valueHard

An auditor draws $5$ trades without replacement from $15$ , of which $6$ are flagged. Let $N$ be the number of flagged trades drawn. What is $Var (N)$ ? Give a fraction in lowest terms.

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Hint

Use indicators for the draws and add the covariances over all ordered pairs of draws.

Solution

Each draw is flagged with chance $\frac{6}{15} = \frac{2}{5}$ , so each indicator has variance $\frac{2}{5} \times \frac{3}{5} = \frac{6}{25}$ .
Both draws of a pair are flagged with chance $\frac{6}{15} \times \frac{5}{14} = \frac{1}{7}$ , so the covariance is $\frac{1}{7} - \frac{4}{25} = - \frac{3}{175}$ .
There are $5 \times 4 = 20$ ordered pairs, so $Var (N) = 5 \times \frac{6}{25} + 20 \times (- \frac{3}{175}) = \frac{6}{7}$ .

Answer: $\frac{6}{7}$

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