FRM Part I · FRM Exam Part I · Sample Moments
Four paired observations of (X, Y) are (2, 1), (4, 5), (6, 3) and (8, 7). What is the unbiased sample covariance of X and Y?
The unbiased sample covariance is 5.33. The cross-products of deviations from the means (5 and 4) sum to 16, and dividing by n minus 1, which is 3, gives 5.33. Dividing by n would give 4.00, which is not the unbiased sample estimator.
- A5.33Correct
- B4.00
- C16.00
- D1.33
Explanation
Means are 5 for X and 4 for Y. Deviation products are (-3)(-3)=9, (-1)(1)=-1, (1)(-1)=-1, (3)(3)=9, summing to 16. The unbiased sample covariance divides by n-1 = 3, giving 5.33. Dividing by n gives 4.00, which is the population-style estimate.
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