Weighting a fixed sum

Let XX, YY, ZZ be independent and identically distributed random variables. Given that X+Y+Z=12X + Y + Z = 12, compute E[2X+3Y]E[2X + 3Y].

Show hints (2)+
  1. By symmetry each of X,Y,ZX,Y,Z has the same conditional mean given their sum — what is it?
  2. Then apply linearity to the weighted combination 2X+3Y2X+3Y.

Answer

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20

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Asked at: Jane Street, Two Sigma

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