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Pricing a squared payoff

MediumExpected ValueMarket Making & Betting Games~4m

A game rolls two fair dice and pays you the square of their sum, in dollars (a 3 and a 5 sum to 8 and pay 82=648^2 = 64 dollars). What is the fair price to play - the expected payout, in dollars to four decimals?

Show hints (2)+
  1. E[S2]≠(E[S])2E[S^2] \ne (E[S])^2 - use E[S2]=Var⁡(S)+(E[S])2E[S^2] = \operatorname{Var}(S) + (E[S])^2.
  2. One die has variance 35/1235/12; independent variances add across the two dice.

Answer

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54.8333 (± 0.001)

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