Statistics 108Homework Assignment 3Note, all problems listed here are to be written up and handed in on the due dateprovided. Homework is posted concurrently with the class material. Sections maybe assigned at different times but have the same due date as previously assignedproblems.Chapter 5 problems below have a due date of May 24(P1) Suppose we have matricesFind (A + B), (A B), (A + B)�, the transpose of the sum, (A B)�, the transposeof the difference.(P2) Consider the matrices�(P3) Consider the matrices�. Compare the results to your answers in problem 2. What doyou notice?(P4) Suppose we have the matrixFind the products A · I where I is the identity matrix and A · J where J is the unitmatrix. Both I and J are 3 × 3 matrices.(P5) Suppose Y is a random vector with elements (Y1, Y2) and X = (X1, X2).(P6) For the random vector Y in problem 5, suppose the variance-covariance matrix�If we let W = A0Y we can show that the variance of W is given by A0ΣY A. Find thevariance ΣW ? What is the dimension of ΣW ?(P7) Consider the matrices A and B and their inverses given below:Verify that A1 and B1 are in fact the inverses. Find (A · B)1 by direct calculationand by using the formula given in chapter 5 on page 7.Chapter 6 problems below have a due date of May 24(P1) Consider the following regression problem: E[Y ] = 2 + 3X with �i ～ N(0, 4) and�independent for i, j = 1, 2, 3, 4. Write the joint distribution of �1, �2, �3, �4. Note,the determinant of Σis given by | Σ|= σ11 · σ22 · σ33 · σ44. Also, note, here σii is tStatistics 108作业代写、代做Python/C++课程作业、代写Java编程设计作业 代写R语言编程|代写Whesame as σ2ii in chapter 6 page 1.(P2) Consider the following regression problem: E[Y ] = 2 + 3X with �i ～ N(0, 4) and�independent for i, j = 1, 2, 3, 4. Write the joint distribution of Y1, Y2, Y3, Y4 forX = 2, 4, 6, 8. Based on your results in problem 1 you only need to calculate theexpression in the exponent of the distribution for Y1, Y2, Y3, Y4.(P3) Suppose, we have the following data: (Xi, Yi) = (88, 20); (70, 26); (72, 28); (64, 23)where X is the percent of households (in a country) with high speed internet accessand Y is the amount of time in hours/week online. For this data write the linearregression model in matrix notation and provide the vector Y as well as the matrixX.(P4) For the data in problem 3 calculate the following quantities: (X0X). Write the normalequations in matrix notation with explicit expressions for the quantities involved.(P5) For the data in problem 3 calculate (X0X)1 using the following formula for theinverse of a 2 × 2 matrix�(a11 · a22 a12 · a21)�Furthermore, calculate the least squares estimates b0 and b1.(P6) For the data in problem 3, calculate the hat matrix and the predicted values andresiduals using the hat matrix.(P7) Consider the data (X1i, X2i, Yi) for the following values(5.9, 3, 6.1); (3.5, 1.9, 5.1); (2.5, 2.9, 4.6); (5.2, 2.8, 5.4); (4.8, 4.6, 6.4);(3.9, 3.3, 4.5); (5.5, 1.7, 4.8); (6.3, 3.3, 5.8); (6.6, 2.7, 5.4)Write down the design matrix (X matrix) for the following models(a) Yi = β0 + β1X1i + β2X2i + �i(b) Yi = β0 + β1X1i + β2X2i + β12X1iX2i + β3X21i + β4X2�转自：http://www.7daixie.com/2019052140782381.html
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