mean and variance of sample median

probability

Solution 1:

  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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