We had a discussion about this question before Thanksgiving break. This problem formalizes that discussion. Let...

 

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We had a discussion about this question before Thanksgiving break. This problem formalizes that discussion. Let X and Y be random variables with E[X] = E[Y] = 0 and Var(X) = Var(Y) = 1. Suppose that the correlation between X and Y is 1. Show that X=Y with probability one. What can we say if Var(X), Var(Y) > 0 but are not necessarily equal to one and E[X] and E[Y] are not necessarily equal to zero? We had a discussion about this question before Thanksgiving break. This problem formalizes that discussion. Let X and Y be random variables with E[X] = E[Y] = 0 and Var(X) = Var(Y) = 1. Suppose that the correlation between X and Y is 1. Show that X=Y with probability one. What can we say if Var(X), Var(Y) > 0 but are not necessarily equal to one and E[X] and E[Y] are not necessarily equal to zero?

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