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Effective sample size

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An importance sampling run has $5$ draws with weights $1$ , $2$ , $3$ , $4$ , $10$ . The effective sample size is the squared sum of the weights divided by the sum of their squares. What is it? Answer as a fraction in lowest terms.

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Hint

Add the weights and square the total, then add the squares of the weights.

Solution

The weights sum to $1 + 2 + 3 + 4 + 10 = 20$ , and the squared sum is $2 0^{2} = 400$ .
The squares sum to $1 + 4 + 9 + 16 + 100 = 130$ .
The effective sample size is $\frac{400}{130} = \frac{40}{13}$ , so the answer is $\frac{40}{13}$ .

Answer: $\frac{40}{13}$

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