Answer to Question #238916 in Statistics and Probability for hamza

Question

Let G1 and G2 be independent Gaussian random variables with mean zero and variance one (i.e. standard

normal r.v.s). Define the random vector X = (X1, X2)^T by X1 = G1 and X2 = G1 + G2. Define Y =

X1 + (X2)^2 + Z, where Z is a uniform random variable on [−1, 1] that is independent of both G1 and G2.

Find the best linear model for predicting the output Y from the input X. In other words, find the vector

β = (β1, β2)^T

that minimizes E((Y − hX, βi)^2

). (Note: Thus, we know that the true regression function is

not linear, but we are still attempting to approximate it by a linear function.)