2 Random variables, X, Y, etc. form a vector space (i.e. they satisfy properties such as closure under addition and scalar multiplication). Furthermore we can define an inner product between random variables as (X,Y) = E[XY] (i.e. the expectation of the random variable Z = XY) (a) Use the Cauchy-Schwarz inequality to show that Cov(X,Y)2 < Var(X) Var(Y) hence show that the Pearson correlation is between -1 and 1.

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