A stochastic signal S is amplified by an amplifier that has a stochastic, real-valued gain, A, so that the amplifiers output is Y = AS. Assume that the random variables A and S are independent. It is known that the random gain is either 3 or 7, with P{A = 3} = 0.8. a) Characterize the transformation g: IR→IR that can be applied to the output Y in or- der to estimate the unknown signal S from the data Y such that the mean-square error, E[g(Y) S²], is minimized. b) Find an explicit formula for such optimal estimator g that you characterized in Part a. c) Calculate the mean-square error assuming that S is a {-1, 1, }-valued random variable with P{S=1}=0.6. d) Now someone has used intuition to guess an answer for Part a as 92(Y) = 5Y/31. Ex- plain where this guessed answer may have come from and comment on the optimality of the guessed estimator 92. e) Calculate the mean-square error for the estimator 92 in Part d.

Consider the linear model y=B,+B,x,+B,x,+B,x+P+u, You estimate the model y=B,+B,x,+B,x, observations and obtain the OLS residuals . You then estimate the auxiliary regression +u based on 123 31 3 31 44i The LM statistic 3i +1 4/ you obtain to test the null hypothesis that H:B,=B,=0 is 20.91. What is the R2 of the auxiliary regression? It is not possible to say 0.17 O 0.175714 O 0.177203

4.19. Consider a regression through the origin, Yi = Bx; +€;, with E(e;)=0, V(€) =o²x}, i=1,2,., 12 a. Derive the generalized least squares estmate of ß in Eq. (4.58) and obtain its variance. Note that the covariance matrix V and its inverse V' are diagonal matrices. The generalized least squares estimate minimizes a weighted sum of squares with weights given by the diagonal elements in V-. Hence, one refers to it as the weighted least squares estimate. b. Suppose that z, = y;/x; and ", Z = 30. Find the numerical value for the weighted least squares estimate in (a) and express its variance as a function of o². %3D

Answer the following questions a, b, c, d with true or false We consider an additive noise model Y-μ(₁) +€₁. where €1,..., ₁ are independent and identically distributed with mean 0 and variance o² > 0. Determine whether each of the following parameterized mean functions (.) leads to a linear model for the nean vector, assuming that the design consists of the points ₁1, ₂2,...,10-10. Note that 3 € RP where p varies for the examples below. Answer true or false to whether each of the mean functions in a.-d. below leads to a linear model. a. . (a) Bo +9₁-2 - b. Ha(z) 8.³ C. ABarBi (2)= Bo+1(31 · >5) d. 8., (2) Bo + ₁ z 1(5)