Bias, variance, MSE, and consistency

Problem

For i.i.d. observations with mean $\mu$ and variance $\sigma^2$ , analyze the sample mean $\overline{X}_n$ as an estimator of $\mu$ : give its bias, variance, MSE, standard error, and whether it is consistent.

Reveal answer or reference solution

The bias is $0$ ; the variance and MSE are $\sigma^2/n$ ; and the standard error is $\sigma/\sqrt{n}$ . It is consistent because the variance tends to zero and the estimator is unbiased.

Local history

Loading attempts saved in this browser…

Use with your agent

Share this URL and your attempt. Ask the agent to start with a clarifying question or the smallest useful hint.

Tutor me on https://mlprep.iwase.dev/mathematics/statistics/original-stats-estimator/. If window.mlPrepAgent is available, read attempts for item original-stats-estimator before tutoring. Inspect my attempt, keep the item ID, and do not reveal the full answer first. After a real attempt, append its record and read it back.

Appears in

Mathematics Diagnostic
Statistics Practice