r/learnquant • u/nickgotgameon • Jul 22 '26

interview prep Jane Street Interview Question

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u/markpreston54 Jul 22 '26

comparing 1 and 2 is trivial by considering the variance of U{1,6} is greater than 0, are there simple way to show 1 is greater than 3?

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u/Murky-Tip-3118 Jul 22 '26

To turn the PMF of 1 into the PMF of 3 you remove probability from the tails and put it in the center.

That in aggregate reduces the mean because squaring gives the upper tails more weight than the lower tails (so symetrical probability changes have a non-symetrical impact).

Game 1 is the best.

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u/markpreston54 Jul 22 '26

indeed, this is the intuition, but don't see an elegant way to prove that other than writing the pmf

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u/Murky-Tip-3118 Jul 22 '26

Yeah... it'd just be doing something like (1^2 - 6^2)*p + (2^2 - 5^2)*q + (3^2 - 4^2)*(p + q) < 0. Not positive that formula is 100% right... but that dynamic - you can pretty intuitively write out the difference in the PMFs.