Assume a random vector x E R follows a multivariate Gaussian distribution (i.e., p(x) N(x ,...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Assume a random vector x E R" follows a multivariate Gaussian distribution (i.e., p(x) N(x μ, E)). If we apply an invertible linear transformation to convert x into another random vector as y = Ax+ b (A € Rn×¹ and b € R"), prove that the joint distribution p(y) is also a multivariate Gaussian distribution, and compute its mean vector and covariance matrix. Show that mutual information satisfies the following: I(X,Y) H(X) – H(XY) H(Y) - H(YX) H(X) + H(Y) - H(X,Y). = = = Compute the distance of a point x0 € IR¹¹ to b) a unit ball ||x|| ≤ 1; c) an elliptic surface x¹ Ax = 1, where A E R"X" and A > 0. Hints: Give a numerical procedure if no closed-form solution exists. Assume a random vector x E R" follows a multivariate Gaussian distribution (i.e., p(x) N(x μ, E)). If we apply an invertible linear transformation to convert x into another random vector as y = Ax+ b (A € Rn×¹ and b € R"), prove that the joint distribution p(y) is also a multivariate Gaussian distribution, and compute its mean vector and covariance matrix. Show that mutual information satisfies the following: I(X,Y) H(X) – H(XY) H(Y) - H(YX) H(X) + H(Y) - H(X,Y). = = = Compute the distance of a point x0 € IR¹¹ to b) a unit ball ||x|| ≤ 1; c) an elliptic surface x¹ Ax = 1, where A E R"X" and A > 0. Hints: Give a numerical procedure if no closed-form solution exists.
Expert Answer:
Answer rating: 100% (QA)
lets break it down step by step 1 Proving Gaussian Transformation Step 11 Given A random vector xRn ... View the full answer
Related Book For
Introduction to Algorithms
ISBN: 978-0262033848
3rd edition
Authors: Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest
Posted Date:
Students also viewed these programming questions
-
Two urns contain k balls each. Initially, the balls in the left urn are all red and the balls in the right urn are all blue. At each step, pick a ball at random from each urn and exchange them. Let X...
-
In this problem, we shall outline a form of robust linear regression. Assume throughout the exercise that the data consist of pairs (Yi, xi) for i = 1, . . . , n. Assume also that the xi s are all...
-
Disturbed, Inc. had the following operating results for the past year: sales = $22,616; depreciation = $1,480; interest expense = $1,192; costs = $16,575. The tax rate for the year was 38 percent....
-
Perform the indicated operations and simplify the result. Leave your answer in factored form. X x - 3 x + 1 x2 + 5x 24
-
A fuel, Cx Hy, is burned with dry air and the product composition is measured on a dry basis to be: 9.6% CO2, 7.3% O2 and 83.1% N2. Find the fuel composition (x/y) and the percent theoretical air...
-
A rough, inclined plane has a string attached to it at the point C. The string has natural length 1.4m and modulus of elasticity 80N. The string is then attached to a particle, P, of mass 4 kg. P is...
-
Explain the reason the Genetic Information Nondiscrimination Act was enacted.
-
Consider a blending tank that has the same dimensions and nominal flow rates as the storage tank in Exercise 2.13 but incorporates a valve on the outflow line that is used to establish flow rate w3....
-
Consider a 10 year bond with face value $1000. Pays 6% coupon semi annual and has a yeild to maruryity of 7%. How much would the approximate percentagechange in the price of bond if interest rate in...
-
Vierra Popova Ltd now wishes to prepare its cash budget for the second six months of the year. The budgeted income statements for each month of the second half of the year are as follows: The...
-
What is the resultant vector? |B|=2 m |A|=5 m d-A-B O The magnitude of C is 7 m and this vector points in the -y direction. O The magnitude of C is 3 m and this vector points in the -Y direction. O...
-
Explain how one would find the value of a nonconstant growth stock.
-
Think of a past job you have held. List three areas in which you, or some other person in the organization, could have benefited from having information generated by research. What would be the most...
-
Explain what is meant by horizon (terminal) date and horizon (continuing) value.
-
Provide an example of each of the following: Business Source Directories, Articles, Dictionaries and Encyclopedias, Marketing Directories, and Statistics and Reports.
-
What steps are taken to find a stock price using the corporate valuation model?
-
About how many expatriates leave their company within one year of returning? About half. More than half. Very few. Less than half. Which is true of a multinational company? It is decentralized. It...
-
What are some of the various ways to implement an awareness program?
-
When RANDOMIZED-QUICKSORT runs, how many calls are made to the random number generator RANDOM in the worst case? How about in the best case? Give your answer in terms of -notation.
-
Define the gcd function for more than two arguments by the recursive equation gcd (a 0 , a 1 , . . . ,a n ) = gcd (a 0 , gcd (a 1 , a 2 , . . . ,a n ). Show that the gcd function returns the same...
-
Find an optimal parenthesization of a matrix-chain product whose sequence of dimensions is 5, 10, 3, 12, 5, 50, 6.
-
In 2018, Cabell Mapp passed away in Belle Haven, Virginia. Pursuant to his will, a testamentary trust was established. The trust instrument requires that \(\$ 10,000\) a year be paid to the...
-
How much may be contributed annually to a defined contribution Keogh plan?
-
Refer to the previous problem. If, instead of maintaining a traditional IRA, Ceci maintains a \(401(\mathrm{k})\) account, what are the deadlines and amounts associated with Ceci's first two RMDs?...
Study smarter with the SolutionInn App