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The dice is 6-faced fair dice. Which of the following gives you the higher expected value: the square of a singular die roll or the square of the median of three dice roll?

Intuitively, the mean of media is same as the mean of sample and the variance of media is smaller than the variance of sample. Then the square of a singular die roll is higher.

For the same mean, it is easy to compute using the symmetry of distribution. But for the variance, it seems not easy to compute or is there any simple way without computation? Moreover, is it true for any i.i.d sample median?

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  • $\begingroup$ Please edit to include your efforts. $\endgroup$
    – lulu
    Dec 23, 2021 at 16:49
  • $\begingroup$ Yes, I misunderstood your intent by the phrase (median of the three dice roll). I have therefore deleted my answer. For what it's worth, my interpretation although not intended by you, does seem like a reasonable interpretation, given your phrasing. $\endgroup$ Dec 23, 2021 at 19:10

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  1. The probability distribution of the dice is invariant under the distribution $x \mapsto 7 - x$; this will swap the highest and lowest die value and keep the median the same. The median and mean must therefore also be invariant under this transformation and be 3.5.

  2. Numbers like 1 and 6 that are far away from 3.5 are less likely to be the middle number

  3. No, only for symmetric distrubitions.

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