Consider the space of all real-valued random variables with finite-variance. Show that this is a vector...

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Consider the space of all real-valued random variables with finite-variance. Show that this is a vector space, and call it S. Note that for X,Y ES, we may view E[XY] as a function mapping Sx S→ R. Show that this is a valid inner product on this vector space. For any pair of finite variance random variables (X,Y), the conditional expectation E[Y|X] is a function of X that is known to satisfy the following property: for all functions¹ E[(Y - E[Y|X])o(X)] = 0. Using this definition, prove that the mean squared error E[(Y-(X))2] of estimating Y from X is minimized by choosing o(X) = E[Y|X]. I.e., the conditional expectation minimizes the mean squared error of estimation. Hint: Think about how we proved the orthogonality principle without necessarily trying to formally define a subspace. Consider the space of all real-valued random variables with finite-variance. Show that this is a vector space, and call it S. Note that for X,Y ES, we may view E[XY] as a function mapping Sx S→ R. Show that this is a valid inner product on this vector space. For any pair of finite variance random variables (X,Y), the conditional expectation E[Y|X] is a function of X that is known to satisfy the following property: for all functions¹ E[(Y - E[Y|X])o(X)] = 0. Using this definition, prove that the mean squared error E[(Y-(X))2] of estimating Y from X is minimized by choosing o(X) = E[Y|X]. I.e., the conditional expectation minimizes the mean squared error of estimation. Hint: Think about how we proved the orthogonality principle without necessarily trying to formally define a subspace.

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To show that the space of all realvalued random variables with finitevariance denoted as S is a vect... View the full answer