The smaller face

Topic: Random variables and distributions · more questions on this topic
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Two fair dice are thrown and Z is the smaller of the two faces.

  1. Give the probability mass function of Z in a table.

  2. Calculate \mathbb{E} Z.

  1. There are 36 equally likely ordered pairs. The number of pairs whose minimum is z is 11,9,7,5,3,1 for z=1,\ldots,6: the pairs in which both faces are at least z, minus the pairs in which both are at least z+1.
z 1 2 3 4 5 6
\mathbb{P}(Z=z) 11/36 9/36 7/36 5/36 3/36 1/36
  1. \mathbb{E} Z=(1\cdot 11+2\cdot 9+3\cdot 7+4\cdot 5+5\cdot 3+6\cdot 1)/36=91/36.