MathDB

Given an ordered list of

3N

real numbers, we can trim it to form a list of

N

numbers as follows: We divide the list into

N

groups of

3

consecutive numbers, and within each group, discard the highest and lowest numbers, keeping only the median. \\ Consider generating a random number

X

by the following procedure: Start with a list of

3^{2021}

numbers, drawn independently and unfiformly at random between

0

and

1

. Then trim this list as defined above, leaving a list of

3^{2020}

numbers. Then trim again repeatedly until just one number remains; let

X

be this number. Let

\mu

be the expected value of

\left|X-\frac{1}{2} \right|

. Show that

\mu \ge \frac{1}{4}\left(\frac{2}{3} \right)^{2021}.

B6