FRM Part I · FRM Exam Part I · Sample Moments
Which statement about the sample mean of i.i.d. observations is correct?
The sample mean is an unbiased estimator of the population mean for any sample size, because the expectation of an average of i.i.d. observations equals the common mean. Normality is not needed for unbiasedness, and its variance shrinks as sigma squared divided by n.
- AIt is a biased estimator of the population mean in small samples
- BIts variance equals the population variance regardless of sample size
- CIt is an unbiased estimator of the population mean for any sample sizeCorrect
- DIt is unbiased only if the underlying distribution is normal
Explanation
E[sample mean] = mu for any n because expectation is linear and each observation has mean mu. Normality is not required for unbiasedness; it only matters for the exact distribution of the mean. Its variance is sigma^2/n, so it falls with sample size.
Did you get it right without looking?
One question tells you little. A timed set on Sample Moments shows your real accuracy, how long you take and where you lose marks.
More Sample Moments questions
- A risk analyst records five daily returns (in percent): 2, 4, 6, 8, 10. Using the unbiased estimator, what is the sample variance of these r…
- Which statement about the sample mean as an estimator of the population mean for i.i.d. data with finite variance is correct?
- A sample of 100 monthly returns has skewness of 0.4 and kurtosis of 4.0. Using the Jarque-Bera statistic, JB = (n/6)[S^2 + (K-3)^2/4], which…
- Observations X1, X2, X3 are i.i.d. with mean mu and variance 36. Estimator A is the sample mean (X1+X2+X3)/3. Estimator B is (X1 + 2*X2 + 3*…
- An analyst records five monthly returns (in %): 2, 4, 6, 8, 10. She wants an estimate of the variance of the underlying return distribution …
- A sample of 5 returns (in %) is: 2, 4, 4, 4, 6. Its sample skewness, defined as the average cubed deviation divided by the cube of the stand…