Random variables
Random variables with zero correlation are not necessarily independent. Give a simple example.
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Let X be a normally-distributed random variable with
Mean zero. Let Y = X^2. Obviously, X and Y are not independent: knowing X, gives the value of Y.
The covariance of X and Y is Cov(X,Y) = E(XY) - E(X)E(Y) = E(X^3) - 0*E(Y) = E(X^3) = 0,
because the distribution of X is symmetric around zero. correlation r(X,Y) = Cov(X,Y)/Sqrt[Var(X)Var(Y)] = 0, the random variables are not independent, but correlation is zero.
Kramer spends all of his income $270 on two products, soup (S) and on golf balls (G). He always bought 2 golf balls for every 1 cup of soup he consumes. He acquires no additional utility from the other cup of soup unless he as well gets 2 more golf balls a
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