Statistics 108辅导、讲解Python/C++、辅导Java编程设计
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Statistics 108
Homework Assignment 3
Note, all problems listed here are to be written up and handed in on the due date
provided. Homework is posted concurrently with the class material. Sections may
be assigned at different times but have the same due date as previously assigned
problems.
Chapter 5 problems below have a due date of May 24
(P1) Suppose we have matrices
Find (A + B), (A B), (A + B)
, the transpose of the sum, (A B)
, the transpose
of the difference.
(P2) Consider the matrices
(P3) Consider the matrices
. Compare the results to your answers in problem 2. What do
you notice?
(P4) Suppose we have the matrix
Find the products A · I where I is the identity matrix and A · J where J is the unit
matrix. 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 the
variance 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 calculation
and 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 the
same as σ2
ii 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 for
X = 2, 4, 6, 8. Based on your results in problem 1 you only need to calculate the
expression 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 access
and Y is the amount of time in hours/week online. For this data write the linear
regression model in matrix notation and provide the vector Y as well as the matrix
X.
(P4) For the data in problem 3 calculate the following quantities: (X0X). Write the normal
equations 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 the
inverse 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 and
residuals 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 + β3X2
1i + β4X2