A model for a certain population P(t) is given by the initial value problem
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- 2. We would like to fit a linear regression estimate to the dataset {(x®,y@),(x), y),., (x(N), g/N)} with x e RM by minimizing the ordinary least square (OLS) objective function: N M -Συ, .(i) J(w): j=1 Specifically, we solve for each coefficient wk (1< k < M) by deriving an expression of Wk from the critical point J(w) the dataset (x(1), y(1)), (x(2), y(2)), . 0. What is the expression for each wk in terms of … , (x(^), y(N) and w1, , wk-1, Wk+1; *** , WM? .. .. Select one: E, (y() –D,-1,j+k W;x;") i=D1 Wk = =1 O WkLet V = R^n and let {e1, . . . , en} be the standard basis for V.a) Give an example of a subspace of V that has dimension m for each 1 <= m <= n.b) Let U = Span(v1, v2, . . . , vn) where v1 = e1 andvi = e1 + ei for 2 <= i <= n.What is the dimension of U?Question 6: Simulate a data set with 150 observations y; = 40+ 6x + 2x² + 3x³ +€, where €; follows independent normal distribution N(0, 1). a) Perform polynomial regression on your simulated data set and using x, I(x²), 1(x³) as the predictors. Compare the estimated coefficients with the true model and report the R-square. b) Formulate the design matrix of this regression and write down the first two rows of the design matrix based on your data set. c) Perform polynomial regression on your simulated data set and using z, I(x²), 1 (x³), 1 (x¹) as the predictors. Compare the estimated coefficients with the true model.Solve the following differential equations: (D² – 4D + 4)y = x³e2* %3D9. Suppose we modify the production model in Section 1.3 to obtain the following mathe- matical model: Max 10x S.t. ax < 40 x 2 0 where a is the number of hours of production time required for each unit produced. With a = 5, the optimal solution is x = 8. If we have a stochastic model with a = 3, a = 4, a = 5, or a = 6 as the possible values for the number of hours required per unit, what is the optimal value for x? What problems does this stochastic model cause?Kindly answer correctly, Please show the necessary steps 1aHow to solve questions b, c, and d2 Specifically, we see from model selection that it may be desirable to select only a subset from available predictors. Suppose that we apply linear regression to a data set, in which we have n observations and p predictors. In fact, when the number of predictors p increases, the resulting linear function will fit the data better and better. Now consider an extreme case: When p = (n – 1), the linear function will fit all the n observations exactly (with probably the exception of a special math case). Without going to details of math, using 2 or 3 sentences, explain why we can fit all the n observations exactly with (n – 1) predictors.16. A data set for which an exponential model is a good fit was transformed. The least squares linear function the transformed data is given by y = 0.1862x2.8688 Find the parameters of the exponential model of the form y = Aert 17. A data set for which a power model is a good fit was transformed. The least squares linear function the transformed data is given by y = = 0.5386x + 0.2561 Find the parameters of the power model of the form y = Ax".