Question

You run 3 miles every day. On a given day, you record the time it takes you (in minutes), and this takes values 25, ..., 31, with each one being equally likely. The time it takes you on any day is independent of the time on other days. You used to record your time every day. Now, you just remember how long it took you every day, and then at the end of the week, record your best time from that week. By how much have you improved your expected recorded time?

Answer 1

Solution: The best time will be just an estimator of the expected value but if we take the average of last week's time and keep increasing the sample size the estimator will tend towards the true mean (unbiased estimator of true mean).

Here all the 7 values from 25 to 31 have the equal probability so the best time for any week will be 25 minutes. If you take mean value as the expected value (i.e. 28 minutes). Hence by 3 minutes you have improved the expected record time.