Lets say we have a lotto game with 6 numbers to choose from (and you can only select different numbers). The expected gain from one ticket is E(x) = -0.8 and a variance of 1000 . Lets say I want to buy multiple tickets for a game and each ticket is independent of eachother meaning I don't know the numbers of the first ticket when buying the second ticket, Does this mean my expected will double ? and what will happen to the variance. And if I do know the first ticket and choose a different combination what will happen to the E(x) and variation ?
Answer
By the linearity of expectation, buying the second ticket doubles the expected value regardless of any correlation in the numbers on the tickets. The change in the variance will depend on whether the tickets agree. If you buy two matching tickets the variance will be multiplied by four. If you buy two tickets that are different enough that they cannot both win any prize simultaneously, the expected value of the square of your winnings doubles while the square of the expected mean is multiplied by four. As the mean is so small compared to the variance, the variance would about double.
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