Multivariate Definition and 135 Threads

Homework Statement Let h(u,v) = f(u+v, u-v). Show that f_{xx} - f_{yy} = h_{uv} and f_{xx} + f_{yy} = \frac12(h_{uu}+h_{vv}) . Homework Equations The Attempt at a Solution I'm always confused on how to tackle these types of questions because there isn't an actual function to...

Homework Statement Let h(u,v) = f(a(u,v), b(u,v)), where a_u = b_v and a_v = -b_u. Show that h_{uu} + h_{vv} = (f_{xx} + f_{yy}) (a^2_u + a^2_v). Homework Equations The Attempt at a Solution I suppose my first question is where the x's and y's come from. (I thought at first it...

Just started multivariate course, can't figure out this simple question. If f(u,v,w) is a function of 3 variables. And u, v and w are themselves function of t. Then does f(u,v,w)=0 implies df/dt=0 or df/du=0. or both.

Hi, I'm an economics graduate student doing some work on a nested logit model. I am trying to generate random variables that follow the following CDF: F(x_1, x_2) =\textrm{exp}[ -(e^{-2x_1}+e^{-2x_2}) ^{1/2}] (This is an extreme-value distribution) With a single random variable, I...

So I figured out the solution to lim (x,y)-->(0,y) of Sin(xy)/x, but I figured it out by looking at a solution. I wanted to understand though why with respect to the above limit how Sin(xy)/x = y? How do you separate the xy in the numerator?

Homework Statement An urn contains $p$ black balls, $q$ white balls, and $r$ red balls; and $n$ balls are chosen without replacement. a. Find the joint distribution of the numbers of black, red, and white balls in the sample. b. Find the joint distribution of the numbers of black and white...

Homework Statement I am trying to find the resultant Gaussian distribution when two multivariate Gaussians are multiplied together - i.e. find the resultant Fisher matrix and mean.Homework Equations Let the two distributions be P_1(x) = \frac{|A|^{0.5}}{(2\pi)^\frac{n}{2}} exp (-0.5 (x-a)^T A...

Hi all, I need help regarding the following expression: E(x1|X>K) where: x1 is a one dimension normal rv X is multivariate normal rv with n components: x1, x2,..., x_n K is a n dimension constant vector with n components: k1,k2,...,k_n X>K <==> x1>k1,x2>k2,...,x_n>k_n I know there is...

Homework Statement Show that : (x^4+y^4)/(x^2+y^2) < ε if 0 < x^2 + y^2 < δ^2 for a suitably chosen δ depending on ε. Homework Equations \forallε>0, \existsδ>0 | 0 < (x^2 + y^2)^(1/2) < δ \Rightarrow |f(x,y) - L| < ε Obviously here were dealing with lim (x,y)→(0,0) f(x,y) = 0 so the...

Hi! I'm trying to linearize a function f which is dependent of 4 variables, each one dependent of time. f[var1_,var2_,var3_,var4_]:= ... expression I use Series[f, {var1,var10 ,1},{var2,var20 ,1},...] syntax as I read on the documentation center. The problem is that the program...

Homework Statement The Attempt at a Solution So I was wondering whether or not, in an instance of n observations and k explanatory variables, where the following is an accurate statement: That is, the estimate of beta_1 found by only regressing y on x_1 is equal to the the true multiple...

Homework Statement Given X,Y RV both have normal distribution with: μ_{x}=6,μ_{y}=4,σ_{x}=1,σ_{y}=5,ρ=0.1 a. are X,Y independent? b. find P(X≤5) c. find P(Y≤5|X=5) 2. The attempt at a solutiona. no -> ρ=0.1 -> cov(X,Y)≠0 b. define Z=\frac{X-6}{1} ; Z~N(0,1) so P(X≤5) = P(Z≤-1) = \Phi(-1)...

Homework Statement Consider f(x,y) = \frac{1}{x} - \frac{1}{y} You need to maximize f(x,y) given the constraint: x + y = 11 Homework Equations I have never solved a problem like this before. In fact I made up a problem a few minutes ago. The Attempt at a Solution I...

Im working my way through a thesis paper that has a lot to do with this. I have a strong background in linear algebra which I've noticed is important as the determinant of the covariance matrix comes up a lot in density functions. Other than the linear algebra though, I've only taken an...

A firm has the following total-cost and demand functions: C = aQ^3 - bQ^2 + cQ + d Q = e - P (d) Find optimizing level of Q. (e) Chooses a,b,c,d and e such that there is only one profit-maximizing level of output Q. I found 2 solutions for question d, but in a very long and messy...

I have two multivariate normally i.i.d random variables, x and y, that are size n vectors. Let us assume for simplicity that their variances are 1. From these random variables, I form two vectors that contain their means, and denote these mx and my. I know that if mx = my, then W = (mx -...

Homework Statement I worked out A) just fine it seems (given the answer in the book), but B) I'm not sure how to take this out. Below was a try but I'm not sure i was even on the right track. Any ideas? Homework Equations The Attempt at a Solution

Hi, I'm new to posting questions on forums, so I apologise if the problem is poorly described. My problem is solving a simulation of the state of a mineral processing froth flotation plant. In the form x@i+1 = f(x@i), f represents the flotation plant. f is a computationally intensive solution...

I have a mixture of multivariate normal distributions, and I want to compute the integral with the first element of the input vector varying between specified limits, and the other elements varying from -infinity to +infinity. See attached pdf for equations. I've done it numerically but would...

It has been awhile since I've posted. :rolleyes: Anyways... I'd like to start learning multivariate calculus along with linear algebra. However, I have a little problem in choosing a suitable text. I'd like to have a text with more theory and less applications. Some people have recommend...

Homework Statement This is part of a larger problem, but in order to take what I believe is the first step, I need to take the Taylor series expansion of f(x,y) = \cos\sqrt{x+y} about (x,y) = (0,0) On the other hand, the purpose of doing this expansion is to find an asymptotic expression for...

Homework Statement We define the function f: \mathbb{R}^2 \to \mathbb{R} as \begin{equation} f(x,y) = \frac{xy^2 ln(x^2 + y^2)}{x^2 + y^2} \end{equation} if (x,y) \neq (0,0). Also note that f(0,0) = 0. Show that f is continuous at (0,0) Homework Equations The Attempt at a...

Homework Statement Let \begin{equation*} f(x,y) = \begin{cases} \dfrac{x^3 - y^3}{x^2 + y^2}, \hspace{1.1em} (x, y) \neq (0,0) \\ 0, \hspace{4em} (x,y) = (0,0) \end{cases} \end{equation*} Is f continuous at the point (0,0)? Are f_x og f_y continuous at the point (0,0)? Homework Equations...

Homework Statement Hi, I have been given a set of functions for which I need to find the stationary points , and determine whether the points are saddle, or max/min. I think I may have solved it correctly but I end up with all the points being saddle, surely this can't be right.. I may...

Homework Statement Show that for all points (\bf{a,b}) from \mathbb{R}^n \times \mathbb{R}^n this applies: \displaystyle \lim _{(\bf{x,y}) \to ({\bf a,b})} \bf{||x-y|| = ||a - b||} Homework Equations Not sure. The Attempt at a Solution I thought about defining a and b as centers of...

Hello, I am working on a problem in which I first sample two unknown probabilities, p and q, from Beta distributions, and then I want to sample both of them at the same time from a multivariate Beta distribution. This is the code that I have for sampling p and q individually from Beta...

Homework Statement 1) A point (X1,X2,X3) are is chosen at random. X1-3 are uniform distributions across the interval [0,1] Determine P[(X1-.5)2 + X2-.5)2 + X3-.5)2 ≤ .25] 2) X and Y are random variables with the joint pdf: f(x,y) = 2(x + y) for 0≤x≤y≤1, 0 otherwise. Find the pdf of Z = X + Y...

Quick Question When taking multiple integrals of a multivariable function, does the order in which you integrate (in terms of the variable) matter? Also, is there a notation for partially integrating a multivariable function with respect to a single variable? Thanks for your help

I urgently need some help in my problem for my MS thesis. I have two datasets of same variable dimension but different number of observations, ie same # of columns but not same # rows. The variables are indentical for both sets. I want to compare the multivariate distributions of the two data...

Homework Statement On average, 2.5 telephone calls per minute are received at a corporation's switchboard. Making appropriate assumptions about the distribution ( provide justification ), find the probability that at any given minute there will be more than 2 calls. Homework Equations No...

Hello all, I wonder if anybody knows of a way of generating a random normal vector (i.e. a variate from a multivariate normal distribituion) in which one or more of the vector's values are fixed?. For example, I may want to choose a random vector from a four-dimensional multivariate normal...

Homework Statement For what real values of the parameters a,b,c,d does the functiob f(x,y)=ax^3+by^3+cx^4+dy^4-(x+y)^5 have a local minimum at (0,0)Homework Equations I calculated the gradient at (0,0) and it is always zero regardless of parameters. The problem is that the Hessian matrix is...

Consider the space of all polynomials in n variables of degree at most d. The dimension of that space is C(n+d,d). How do I calculate the dimension of that same space when I restrict the domain of the polynomials to the unit ball? In that case all the polynomials (sum(i=1..n) x_i^2)^p with p a...

hi guys, I'm trying to understand how the critical value of Q-statistics is calculated in multivariate ljung-box test. or even more simply, in the univariate ljung-box test. initially i used MATLAB for hypothesis testing, MATLAB as an output furnishes the critical values, but the modules in...

Fermat's theorem provides that, if a function f(x) has a local max or min at a, and if f'(a) exists, then f'(a)=0. I was wondering whether a similar theory exists for a function f(x,y) or f(x,y,z) etc.

How can I prove that the any linear combination of multivariate normal distribution is also normal? I can prove it but I'm not sure that this is right or not. The point of my proof is as follows. --- The X and Y has the same dimensional random vector, and each random vector is...

The fisher information matrix for multivariate normal distribution is said at many places to be simplified as: \mathcal{I}_{m,n} = \frac{\partial \mu^\mathrm{T}}{\partial \theta_m} \Sigma^{-1} \frac{\partial \mu}{\partial \theta_n}.\ even on...

So if I start with a multivariate distribution f(x,y), I can find the marginal distributions, the conditional probability distributions, all conditional moments, and by the law of iterated expectations, the moments of both X and Y. It seems to me that I should be able to relate the conditional...

How is hypothesis testing performed for multivariate data? Say for simplicity we have two iid draws from a binomial distribution Bin(10,q) with X1=7, X2=8. Under the null hypothesis H0:q=1/2, the individual p-values (as one-tail probabilities) are approximately 0.172 and 0.055 respectively...

lim (xy^2)/(x + y^2) (x,y)->(0,0) i just came up with this limit to test my understanding. so approaching from y = mx, y = mx^2, and y = mx^3 all seemed to point out that the limit is 0. i tried approaching from the x and y axes and i got for approaching along the x axis: lim (as...

Homework Statement Hello, If you don’t mind, would you all be able to look over these problems for me? I have left out much of the ‘algebra’ work for the sake of space…so please let me know if you think there are any algebra errors. In addition I am particularly looking to see if I have set...

Homework Statement Consider the surface x^2 / 4 + y^2 + z^2 = 3 (a) what type of quadric surface is this? (a spheroid...) (b) find the outward unit normal vector to this surface at the point (2,1,1). Homework Equations How do we find a normal vector? Does one just take grad(f(2,1,1))...

Homework Statement Apply the two cases of th change rule. For example: The voltage V in an electrical circuit is slowly decreasing as a battery wears out and the resistance R is slowly increasing as the resistor heats up Use Ohm's law V=IR to find how the current is changing (with respect to...

Hi all, I'm working towards my Msc dissertation and I've ran into a tricky problem which I've figured it out to be modeled as the following urn problem: there are m balls and m urns U_1, ..., U_m with capacities C(U_1) = m, C(U_2) = m-1, ..., C(U_m) = 1. Knowing that each urn U_i is only...

Homework Statement I am supposed to show whether the limit: lim (x,y)----> (0,0) of yln(x^2 + y^2) exists or doesn't. Homework Equations The Attempt at a Solution I've tried numerous paths, but what it seems to come down to is showing that the y factor goes to zero quicker...

I've spent a lot of time soul searching after having some academic "failures" in the previous semesters, and what I found is that often when I'm not studying, it's not because I can't study, but it's because the book we're using is unreadable to me. For example, after sailing through Calculus...

I wish to master Multivariate Calculus(to the extent required to get an A with a ridiculously difficult professor) and Linear Algebra in 3 weeks so I can breeze through both classes in the Fall semester. I've self studied both of my textbooks that I will be using for the two courses, but I'd...

Homework Statement Find \frac{\partial z}{\partial y} [/itex] where z=F(u,v,y), u=f(v,x), v=g(x,y). The Attempt at a Solution If I remember multivariate calculus at all, this should be (please forgive the abuse of notation) \frac{\partial z}{\partial y} = \frac{\partial z}{\partial...

Hi. I am going to have my Multivariate Statistics exam in two weeks. I am going by a book that isn't available in the English literature. But I don't like it, because it has several omissions. Can you recommend me any good books in English? I can borrow many (for free) at my institution's...

The standard multivariate gaussian is given by: http://upload.wikimedia.org/math/1/c/d/1cd250fc27ef7b7a9da469416333d07f.png taken from: http://en.wikipedia.org/wiki/Multivariate_normal_distribution The parameters can be estimated using...