3 Multivariate Gaussian distributions A multivariate Gaussian distribution has a density function defined as 1 p(x)...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
3 Multivariate Gaussian distributions A multivariate Gaussian distribution has a density function defined as 1 p(x) = = exp(-²/(r- (x − *) Σ-1(x – - â)). Here, & ER" is the mean of the distribution, and Σ => 0 is its covariance matrix, and the value C is such that [p(x) dx = 1. 1. (2 points) Show that when â = 0 and Σ = I, the constant C takes the value C = (2n)"/2. Hint: The integral of a product of functions fi, each depending on a single variable zi, can be expressed as a product of n one-dimensional integrals: n [ ÎI fi(zi) dz = Ï _ fi(ži) dzi. i=1 i=12₁ER Note that when calculating the individual integrals, you may use the 1-d result for standard normal distribution without proof. 2. (2 points) Using a spectral decomposition of Σ, show that in general, for arbitrary and matrix > 0, we have C = (2π)"/2√det Σ. Hint: for any integrable function f and square invertible matrix P, we have f(x) dx = |det P. S.cz, S(P2) dz. ZER" IER" 3. (2 points) Show that is indeed the mean of the distribution, that is: â = B(x) = √xp(x) dx. 4. (2 points) Show that is indeed the covariance matrix of the distribution, that is: (x-2)(xê) p(x) dx. Again you may assume that the result holds true when n = 1. Hint: use an affine change of variables in order to reduce the problem to that of computing the covariance of the distribution obtained upon setting 2 = 0 and Σ = I, which is the matrix Σ= M:= Rn (2m)n/² · ___²2¹e-1²³² d₂. 5. (2 points) Explain how to generate samples from the multivariate distribution, assuming that you have access to a sample generator for the normal distribution, which is the scalar distribution with zero mean and unit variance. ZER 3 Multivariate Gaussian distributions A multivariate Gaussian distribution has a density function defined as 1 p(x) = = exp(-²/(r- (x − *) Σ-1(x – - â)). Here, & ER" is the mean of the distribution, and Σ => 0 is its covariance matrix, and the value C is such that [p(x) dx = 1. 1. (2 points) Show that when â = 0 and Σ = I, the constant C takes the value C = (2n)"/2. Hint: The integral of a product of functions fi, each depending on a single variable zi, can be expressed as a product of n one-dimensional integrals: n [ ÎI fi(zi) dz = Ï _ fi(ži) dzi. i=1 i=12₁ER Note that when calculating the individual integrals, you may use the 1-d result for standard normal distribution without proof. 2. (2 points) Using a spectral decomposition of Σ, show that in general, for arbitrary and matrix > 0, we have C = (2π)"/2√det Σ. Hint: for any integrable function f and square invertible matrix P, we have f(x) dx = |det P. S.cz, S(P2) dz. ZER" IER" 3. (2 points) Show that is indeed the mean of the distribution, that is: â = B(x) = √xp(x) dx. 4. (2 points) Show that is indeed the covariance matrix of the distribution, that is: (x-2)(xê) p(x) dx. Again you may assume that the result holds true when n = 1. Hint: use an affine change of variables in order to reduce the problem to that of computing the covariance of the distribution obtained upon setting 2 = 0 and Σ = I, which is the matrix Σ= M:= Rn (2m)n/² · ___²2¹e-1²³² d₂. 5. (2 points) Explain how to generate samples from the multivariate distribution, assuming that you have access to a sample generator for the normal distribution, which is the scalar distribution with zero mean and unit variance. ZER
Expert Answer:
Related Book For
Posted Date:
Students also viewed these law questions
-
The Rayleigh distribution has probability density function (a) It can be shown that E(X2) = 20. Use this information to construct an unbiased estimator for θ. (b) Find the maximum likelihood...
-
For Review Exercise 6.72, what is the mean of the average water usage per hour in thousands of gallons?
-
Given the results described in Exercise, is the mean of 140.0 executives expressed with m or x? Explain why the mean of 140.0 executives is a parameter instead of a statistic.
-
Classify each of the following random variables as either continuous or discrete: a. The survival time of a cancer patient after receiving a new treatment for cancer b. The number of ticks found on a...
-
On January 1, 2011, Timothy Company purchased Macys Company at a price of $3,750,000. The fair market value of the net assets purchased equals $2,700,000. 1. What is the amount of goodwill that...
-
The loz (lethal on Z) mutants of bacteriophage X are conditional lethal mutants that can grow on E. coli strain Y but cannot grow on E. coli strain Z. The results shown in the following table were...
-
The following contingency table presents observed frequencies. a. Compute the expected frequencies. b. One of the observed frequencies is less than 5. Is it appropriate to perform a test of...
-
Mulberry Sporting Goods is authorized to issue 18,000 shares of common stock . During a two-month period, Mulberry completed these stock-issuance transactions: Apr 23 Issued 4,000 shares of $1.00 par...
-
Give a article in any publication that discusses unusual, or interesting forecasting method or use of forecasting a business operation Then summarize the article you found Provide a link to or...
-
The case of Michigan Basic Property Insurance Association v. Washington (MICHIGAN BASIC PROPERTY INSURANCE ASSOCIATION V. JOYCE WASHINGTON, CLEVELAND POWELL, COUNTRYWIDE HOME LOANS, DIMONTI &...
-
Helsen, inc, manufactures lampshades the company is now preparing detailed budgets for the third quarter and has assembled the following information to assist in the budget preparation. a. The...
-
Find three different ways to add up all the numbers from 1 to 1000, inclusive. First, use the built in sum() function. You should be able to write this in one line.
-
The financial planning process involves the consideration of several strategic factors, including how capital should be allocated, outside market forces, competitive factors ,access to capital...
-
Q1: Write down regular expressions for each of the following languages. Assume = {0,1] [6 marks] The language contains strings where Os and Is always appear in pairs. For example-001100,...
-
The first audit of the books of Teal Company was made for the year ended December 31, 2026. In examining the books, the auditor found that certain items had been overlooked or incorrectly handled in...
-
A boy drags a 100N sled up a 20 slope at constant velocity. If the coefficient of friction between sled and hill is 0.20, what force must he exert at an angle of 35 with respect to the hill? ( 4marks...
-
NiseFoodCorporation is analyzing the possibility to acquire... NiseFoodCorporation is analyzing the possibility to acquire KradFood Corp. Kradfood earnings have been under pressure in recent years,...
-
Making use of the tables of atomic masses, find the velocity with which the products of the reaction B10 (n, ) Li7 come apart; the reaction proceeds via interaction of very slow neutrons with...
-
A coin is flipped 25 times. Let x be the number of flips that result in heads (H). Consider the following rule for deciding whether or not the coin is fair: Judge the coin fair if 8 # x # 17. Judge...
-
USA Today ( May 9, 2006) published the average weekday circulation for the top 20 newspapers in the country. Here are the data for the 6- month period ending March 31, 2006: 2,272,815 2,049,786...
-
The article Should Pregnant Women Move? Linking Risks for Birth Defects with Proximity to Toxic Waste Sites (Chance [1992]: 40 45) reported that in a large study carried out in the state of New York,...
-
A Laplacian operator is given by a. \(\frac{\partial^{2}}{\partial x \partial y}\) b. \(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+2 \frac{\partial^{2}}{\partial x...
-
A cable of length \(l\) and mass \(ho\) per unit length is stretched under a tension \(P\). One end of the cable is connected to a mass \(m\), which can move in a frictionless slot, and the other end...
-
Fill in the Blank. The quantities \(E I\) and \(G J\) are called the ___________ and ____________ stiffnesses, respectively.
Study smarter with the SolutionInn App