(Solution) - Let X1 Xn be a random sample -(2025 Original AI-Free Solution)

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Let X1, . . ., Xn be a random sample from an N (?, ?2) distribution. Show that the sample median, M, is an unbiased estimator of the population mean ?. Compare the variances of X-bar and M. For the normal distribution, the mean, median, and mode all occur at the same location. Even though both X-bar and M are unbiased, the reason we usually use the mean instead of the median as the estimator of m is that X-bar has a smaller variance than M.