Let Z (Z,..., Zp) form a random vector where each variable has a N(0, 1) dis-...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Let Z (Z,..., Zp) form a random vector where each variable has a N(0, 1) dis- tribution. Let be a positive definite symmetric square matrix and = TT be the Cholesky decomposition of where T is an upper triangular matrix. (You may also find an alternative formulation using the lower triangular matrix, i.e., E = LLT. However, for this problem use the upper triangular formation.) i) (10 points) We want to generate observations from N(, E) using the transfor- mation X = TZ+. Show that, if Z,..., Z,~ N(0, 1), then X~ N(1, 2), using Cholesky decom- position of , where ERP and E RPXP. Find E(X) and Var(X). ii) (10 points) Let 4 -1.5 0.8 and E = -1.5 1 -0.1 0.8 -0.1 1 = (002) Using R, generate n = 1000 observations from N3(4, E) using the Cholesky de- composition technique. Do not use the built-in R function. Make a scatterplot matrix to visualize all the bivariate relationships between the variables. [Hint: You can use the R command chol to perform the Cholesky decomposition. You may notice that it outputs an upper triangular matrix.] Let Z (Z,..., Zp) form a random vector where each variable has a N(0, 1) dis- tribution. Let be a positive definite symmetric square matrix and = TT be the Cholesky decomposition of where T is an upper triangular matrix. (You may also find an alternative formulation using the lower triangular matrix, i.e., E = LLT. However, for this problem use the upper triangular formation.) i) (10 points) We want to generate observations from N(, E) using the transfor- mation X = TZ+. Show that, if Z,..., Z,~ N(0, 1), then X~ N(1, 2), using Cholesky decom- position of , where ERP and E RPXP. Find E(X) and Var(X). ii) (10 points) Let 4 -1.5 0.8 and E = -1.5 1 -0.1 0.8 -0.1 1 = (002) Using R, generate n = 1000 observations from N3(4, E) using the Cholesky de- composition technique. Do not use the built-in R function. Make a scatterplot matrix to visualize all the bivariate relationships between the variables. [Hint: You can use the R command chol to perform the Cholesky decomposition. You may notice that it outputs an upper triangular matrix.]
Expert Answer:
Answer rating: 100% (QA)
i To show that X follows a multivariate normal distribution N using the transformation X ... View the full answer
Related Book For
Posted Date:
Students also viewed these mathematics questions
-
Han invests $13,000 into a fund that combines stocks and bonds. The return varies from year to year. The balance at 5- year intervals is given in the table below 0 5 10 15 20 25 y (Balance / $1,000)|...
-
Let A be a symmetric positive definite n à n matrix. (a) If k where yk Rk and βk is a scalar, show that Lk+1 is of the form and determine xk and αk in terms of Lk,...
-
The following additional information is available for the Dr. Ivan and Irene Incisor family from Chapters 1-5. Ivan's grandfather died and left a portfolio of municipal bonds. In 2012, they pay Ivan...
-
A small button placed on a horizontal rotating platform with diameter 0.320 m will revolve with the platform when it is brought up to a speed of 40.0 rev/min, provided the button is no more than...
-
Briefly discuss the transactional approach to income measurement. Explain its relationship (if any) to the capital maintenance concept of income.
-
The price of aspirin rose sharply last month, while the quantity sold remained the same. Five people suggest various diagnoses of the phenomenon: Meredith: Demand increased, but supply was perfectly...
-
Why do we need a theory to understand real-world labor market problems?
-
Refer to the Buena School District bus data. a. Develop a 95% confidence interval for the mean bus maintenance. b. Develop a 95% confidence interval for the mean bus miles.
-
The Shirt Works sells a large variety of tee shirts and sweatshirts. Steve Hooper, the owner, is thinking of expanding his sales by hiring high school students, on a commission basis, to sell...
-
1. Based on the marketing activities DNBC has tried, do you think it has a production-oriented marketing or a consumer-oriented philosophy? Explain your reasoning with examples. 2. Research the City...
-
You measure a sound level (at your location) of approximately 109 dB from a firework exploding 25 m away from you. Assuming that your ear has an effective radius of about 2.8 cm, what is the total...
-
Visit the website of Alpha Analytics (www.alpha-analytics.com/home). Briefly describe their data cleaning and validation techniques and how they are applied.
-
An electron is released from rest in a uniform electric field of magnitude \(4.0 \times 10^{3} \mathrm{~N} / \mathrm{C}\). What is the electrostatic work done by the electric field on the particle...
-
Search the Internet including social media to obtain information on flying and passengers' airline preferences. Develop three double-barreled questions related to flying and passengers' airline...
-
A particle of energy E traveling from the left hits a barrier of height U > E and thickness L. Calculate the transmission coefficient. Student can find a simulation of this problem at...
-
Develop the questionnaire in Internet Exercise 8-21 using an electronic questionnaire design package that is available for free (see the websites mentioned in this chapter in the subsection entitled...
-
Solve (8x + 5y)dx + (5x + 20y)dy = 0 Follow the step-by-step instruction below to learn to solve the exact differential equation. Suppose P(x, y) = (8x + 5y) and Q(x, y) = (5x + 20y) P(x, y)dx + Q(x,...
-
When the concentration of a strong acid is not substantially higher than 1.0 10-7 M, the ionization of water must be taken into account in the calculation of the solution's pH. (a) Derive an...
-
Consider the matrices Note that the two matrices are the same except for their (2, 2) entries. (a) Use MATLAB to compute the eigenvalues of A and B. Do they have the same type of eigenvalues? The...
-
Prove that if A is row equivalent to B, then B is row equivalent to A.
-
Let A be a matrix whose columns all add up to a fixed constant . Show that is an elgenvalue of A.
-
A dimensionless grouping of variables and parameters that are important in pipe flow is called the Reynolds number: where $V$ is the average velocity over the cross section in a pipe of diameter $D$...
-
A layer of water is flowing down a flat plate that is inclined at an angle of $20^{\circ}$ to the vertical. If the depth of the layer is $1 / 4 \mathrm{in}$., what is the shear stress exerted by the...
-
Basic Concepts by considering the simple one-dimensional steady flow between two parallel plates, Figure 1.1. The area of each plate $A_{y}=1 \mathrm{~m}^{2}$ and the gap between the two plates is...
Study smarter with the SolutionInn App