Refinement on the first answer to Q1. Actually it also has a lower bound, (a-b)^2. Upper bound is correct, (a+b)^2.

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Quantitative Research Interview Boston, MA (US)

AKUNA CAPITALAnswer## Let X and Y be random variables with standard deviation a

and b. What are the bounds of Var(X + Y)? Pile of 100 coins, 99 fair 1 double heads. Pick random coin, flip 10 times, get 10 heads. What is the probability the coin is double headed coin? Dice sum games.

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Refinement on the first answer to Q1. Actually it also has a lower bound, (a-b)^2. Upper bound is correct, (a+b)^2.

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1. Var(X+Y) = Var(X) + Var(Y) + 2Cov(X, Y) <= Var(X) + Var(Y) + 2*sqrt[Var(X)*Var(Y)] = a^2 + b^2 + 2ab = (a+b)^2

2. Use bayesian theorem:

event A is pick double heads coin

event B is flip 10 times and get 10 heads

Given event A then probability of event B is 1^10, i.e. P(B|A) = 1

P(A|B) = P(B|A)P(A)/P(B) = 1*(1/100) / [1*(1/100) + (1/2)^10 * (99/100)] = 1024/(1024+99) = 1024/1123 = 0.9118