Homework Answers
As nothing is mentioned in the question, it is assumed that X and Y are independent random variables. Also, the first order approximation has been used to calculate Expectation and Variance of functions.
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7. Error Propagation (20 pts) Consider the following statements, where ,4-2 and μ,-1, with σ1-0.02 and...
7. Let W = Span{x1, x2}, where x1 = [1 2 4]" and X2 – [5...
7. Let W = Span{x1, x2}, where x1 = [1 2 4]" and X2 – [5 5 5]" a. (4 pts) Construct an orthogonal basis {V1, V2} for W. b. (4 pts) Compute the orthogonal projection of y = [0 1]' onto W. C. (2 pts) Write a vector V3 such that {V1, V2, V3} is an orthogonal basis for R", where vi and v2 are the vectors computed in (a).
Question 1 4.5 pts Given are five observations for two variables, x and y. 3 2...
Question 1 4.5 pts Given are five observations for two variables, x and y. 3 2 7 h. Compute the mean square error (MSE) O 3.875 4.133 4.421 4.312 i. Compute the standard error of the estimate 2.326 1.895 2.033 2.213 j. Compute the estimated standard deviation (standard error) of bu 0.643 0.697 0.568 O 0.713 k. Test for a significant relationship using the t test. Compute the value of the test statistic and the p-value. At a = 0.05,...
3. (20 pts.) Consider the system: where: C-1 0 0 -4 0 7 a) Analyze the...
3. (20 pts.) Consider the system: where: C-1 0 0 -4 0 7 a) Analyze the stability of the system. b) Design a regulator for this system using the pole-placement with observer approach. Assume that the desired closed-loop poles for the pole placement part are located at: The desired observer poles are: S--7 c) Obtain the transfer function of the observer controller. S--7
4. (Matlal) attatimient) Consider the initial valle probleni 1<t< 2 y(1) 1 Caleulate the approximate solutions...
4. (Matlal) attatimient) Consider the initial valle probleni 1<t< 2 y(1) 1 Caleulate the approximate solutions using forward Euler method, two stage and four stage Runge Kutta method with h 1/10, 1/20,1/40 and compute the maximum errors between the exact solution and the approximate solutions. Use this maximum error to verify the convergence order of each method (1, 2, and 1). Note: the exact solution is
7. (20 pts) Consider the surface given by z wy - 4xy + 3x - 2y....
7. (20 pts) Consider the surface given by z wy - 4xy + 3x - 2y. (a) Find the equation of the tangent plane to this surface at the point where (x,y) - (1,3). (b) Find the gradient f at the point where (x,y) = (1,3). (c) Find the directional derivative DaS(1,3) where it is the unit vector in the direction of (1, -2). such that the directional derivative Daf(1.3) is a maximum (d) Find a unit vector in the...
4. (20 pts) Consider the following regression model, i = 1,2. ,...n, N (0, 2/i) where...
4. (20 pts) Consider the following regression model, i = 1,2. ,...n, N (0, 2/i) where , 1,2,...,n are independent, but c; ~ (a) Do you think if it is suitable for the (ordinary) least square regression technique to apply the data (x4, Y;)? Give a brief reasoning (b) Construct a transformed model so that you can use the ordinary least square method (c) Find the parameter estimates for the transformed model in (b) WIS (d) Find the weighted least...
Part 1/2 Consider the circuit shown in Figure 1, where RD = 1kQ, Vtn-0.9V, μ. Cox...
Part 1/2 Consider the circuit shown in Figure 1, where RD = 1kQ, Vtn-0.9V, μ. Cox (W/L) = 1.5 mA/V and 2 0. Perform DC analysis to compute the voltages Vi and V2. -2.5V Figure 1 Part 2/2 Compute the small-signal voltage gain voVsi. Hints to the solution 1. Redraw the circuit in the AC small-signal domain as shown below, Roy 2. Find the small-signal equivalent Thevenin resistance seen between the source of M2 and the ground node, Re, indicated...
2. (20 pts) Consider the following function with its second derivative in simplified form (you should...
2. (20 pts) Consider the following function with its second derivative in simplified form (you should compute the first derivative): 4.2 f(x) = x²+1 f"(x) = 8.2(x - 3)(x + 3) (x² + 13 "None" could be an appropriate answer for some of the statements below: (a) (2 pts) The domain of f(x) is (b) (2 pts) An x-intercept of the graph of y = f(x) occurs at x = (c) (2 pts) The graph of y = f(x) has...
Question 1 20 pts Determine Oplfor the following element in psi (to the nearest 0.01 psi)...
Question 1 20 pts Determine Oplfor the following element in psi (to the nearest 0.01 psi) 0,4 psi 16 psi Try 24 psi Next > 30 4 + 2 $ 4 % 5 A 6 3 & 7 B W E R T Y S D F H J к
20. (4 pts) Consider the following recurrence. an = 2an-1 + 2an-2 ao = 0 Q1...
20. (4 pts) Consider the following recurrence. an = 2an-1 + 2an-2 ao = 0 Q1 = 2V3 For what values of a and B is the following expression a solution of that recurrence? a;=a (1+ v3)*+B (1 - v3)' a = -1 and B 1 O a = = { and B :- O a = 2 and B = -2 the four other possible answers are incorrect O a = 1 and B = -1
7. Let W = Span{x1, x2}, where x1 = [1 2 4]" and X2 – [5 5 5]" a. (4 pts) Construct an orthogonal basis {V1, V2} for W. b. (4 pts) Compute the orthogonal projection of y = [0 1]' onto W. C. (2 pts) Write a vector V3 such that {V1, V2, V3} is an orthogonal basis for R", where vi and v2 are the vectors computed in (a).
Question 1 4.5 pts Given are five observations for two variables, x and y. 3 2 7 h. Compute the mean square error (MSE) O 3.875 4.133 4.421 4.312 i. Compute the standard error of the estimate 2.326 1.895 2.033 2.213 j. Compute the estimated standard deviation (standard error) of bu 0.643 0.697 0.568 O 0.713 k. Test for a significant relationship using the t test. Compute the value of the test statistic and the p-value. At a = 0.05,...
3. (20 pts.) Consider the system: where: C-1 0 0 -4 0 7 a) Analyze the stability of the system. b) Design a regulator for this system using the pole-placement with observer approach. Assume that the desired closed-loop poles for the pole placement part are located at: The desired observer poles are: S--7 c) Obtain the transfer function of the observer controller. S--7
4. (Matlal) attatimient) Consider the initial valle probleni 1<t< 2 y(1) 1 Caleulate the approximate solutions using forward Euler method, two stage and four stage Runge Kutta method with h 1/10, 1/20,1/40 and compute the maximum errors between the exact solution and the approximate solutions. Use this maximum error to verify the convergence order of each method (1, 2, and 1). Note: the exact solution is
7. (20 pts) Consider the surface given by z wy - 4xy + 3x - 2y. (a) Find the equation of the tangent plane to this surface at the point where (x,y) - (1,3). (b) Find the gradient f at the point where (x,y) = (1,3). (c) Find the directional derivative DaS(1,3) where it is the unit vector in the direction of (1, -2). such that the directional derivative Daf(1.3) is a maximum (d) Find a unit vector in the...
4. (20 pts) Consider the following regression model, i = 1,2. ,...n, N (0, 2/i) where , 1,2,...,n are independent, but c; ~ (a) Do you think if it is suitable for the (ordinary) least square regression technique to apply the data (x4, Y;)? Give a brief reasoning (b) Construct a transformed model so that you can use the ordinary least square method (c) Find the parameter estimates for the transformed model in (b) WIS (d) Find the weighted least...
Part 1/2 Consider the circuit shown in Figure 1, where RD = 1kQ, Vtn-0.9V, μ. Cox (W/L) = 1.5 mA/V and 2 0. Perform DC analysis to compute the voltages Vi and V2. -2.5V Figure 1 Part 2/2 Compute the small-signal voltage gain voVsi. Hints to the solution 1. Redraw the circuit in the AC small-signal domain as shown below, Roy 2. Find the small-signal equivalent Thevenin resistance seen between the source of M2 and the ground node, Re, indicated...
2. (20 pts) Consider the following function with its second derivative in simplified form (you should compute the first derivative): 4.2 f(x) = x²+1 f"(x) = 8.2(x - 3)(x + 3) (x² + 13 "None" could be an appropriate answer for some of the statements below: (a) (2 pts) The domain of f(x) is (b) (2 pts) An x-intercept of the graph of y = f(x) occurs at x = (c) (2 pts) The graph of y = f(x) has...
Question 1 20 pts Determine Oplfor the following element in psi (to the nearest 0.01 psi) 0,4 psi 16 psi Try 24 psi Next > 30 4 + 2 $ 4 % 5 A 6 3 & 7 B W E R T Y S D F H J к
20. (4 pts) Consider the following recurrence. an = 2an-1 + 2an-2 ao = 0 Q1 = 2V3 For what values of a and B is the following expression a solution of that recurrence? a;=a (1+ v3)*+B (1 - v3)' a = -1 and B 1 O a = = { and B :- O a = 2 and B = -2 the four other possible answers are incorrect O a = 1 and B = -1
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