Multivariate Definition and 121 Threads

Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...

Hi, I was attempting the problem above and got stuck along the way. Problem: Suppose that ## Y_1 ## and ## Y_2 ## are random variables with joint pdf: f_{y_1, y_2} (y_1, y_2) = 8y_1 y_2 for ## 0 < y_1 < y_2 < 1 ## and 0 otherwise. Let ## U_1 = Y_1/Y_2 ##. Find the probability distribution ##...

Hello, With multivariate linear regression, there is a single dependent variable ##y## and multiple independent variables ##x_1##, ##x_2##, ##x_3##, etc. There is a linear, weighted relationship between ##y## and the various ##x## variables: $$ y = c_1 x_1 + c_2 x_2 + c_3 x_3 $$ The...

Solution attempt: $$F(F(x,y))=(z,w)$$ is the map given by $$x=z$$ $$y=w$$

First thank you for taking your time to take a look at this simple question. And sorry for the informal math language and equations, I hope you guys can understand it. So, depending on the case, I have 2 or 8 simple quadratic functions f(a), f(b), f(c),… f(z). Each a,b,c,…,z have a different...

Hi all, I am trying to figure out a way to simplify this problem to give the image of the map. I have not seen this function before and I am having trouble figuring out what the image should come out as. I have tried graphing u and v separately as a function of x and y in R3 and both surfaces...

I just finished multivariate calculus (without any linear algebra experience yet) and I am seeking out a path to understanding General Relativity. I am wondering what are the mathematical fields after multivariate calculus that I need to master before beginning to understand GR, and what...

Thank you to all those who helped me solve my last question. This week, I've been assigned an interesting problem about toruses. I think I've solved most of this problem on my own, but I'd like to hear a few suggestions for part c. I think this map multiplies tangent vectors by a factor of...

Any help with this introductory differential geometry HW would be greatly appreciated. My attempt at solving the first problem: y=4x^3-3x has the derivative 12x^2-3, which is 0 when x^2=(1/4). {x^2=(1/4)} is the singular set, and the inverse is defined for everywhere except F({x^2<(1/4)}). the...

For the given joint pdf of X and Y $$f(x,y) = 12xy(1 - y); 0 < x < 1; 0 < y < 1$$ Let $Z = XY^2$ and $W = Y$ be a joint transformation of (X,Y) Sketch the graph of the support of $(Z,W)$ and describe it mathematically. I'm not very sure how to describe (Z,W). First, I draw the graph of the...

1. We find the partial derivatives of ##f## with respect to ##x## and ##y## to get ##f_x = \frac{2\ln{(x)}}{x}## and ##f_y = \frac{2\ln{(y)}}{y}.## This makes the gradient vector $$\nabla{f} = \begin{bmatrix} f_x \\ f_y \end{bmatrix} = \begin{bmatrix} \frac{2\ln{(x)}}{x} \\ \frac{2\ln{(y)}}{y}...

There are sets of the form ##\left\{(x,y)\in \mathbb{R}^2: f(x,y) = \ln{\left(3+(x+y)^2\right)} = c\right\}## where ##c## is some fixed number ##> 1##. Let's see what happens for a few values of ##c##. Suppose ##c = 2##, then ##\ln{\left(3+(x+y)^2\right)} = 2 \Longleftrightarrow (x+y)^2 =...

Hello. I would like to draw (sample) several random vectors x from a n-dimensional multivariate normal distribution. For this purpose I want to use C++ and the GNU Scientific Library function gsl_ran_multivariate_gaussian ...

Homework Statement [1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...

Afternoon, I have been working on this problem for awhile now but have been stuck on a certain point, and once I set everything equal to each other I end up with the same thing for example: x=x, y=y, & z=z Find the minimum and maximum values of the function f(x,y,z)=3x+2y+4z subject to the...

I have been stumped on this problem and I am probably overthinking it as usual. The problem deals with the Lagrange Multipliers: Use Lagrange multipliers to find the point (a,b) on the graph of y=e^(9x), where the value (a,b) is as small as possible. I have found the gradient for both when...

Okay, I never took calculus 3 so I don't know how to write/solve an integral for a power I made up. I want to flesh out my game world with math because it's cool and I can. Like Chester wrote some of the physics for the Bizarro universe. Applying physics and math to a fictitious universe is...

Homework Statement [/B] ##-1\leq\alpha\leq 1## ##f(y_1,y_2)=[1-\alpha\{(1-2e^{-y_1})(1-2e^{-y_2})\}]e^{-y_1-y_2}, 0\leq y_1, 0\leq y_2## and ##0## otherwise. Find ##V(Y_1-Y_2)##. Within what limits would you expect ##Y_1-Y_2## to fall? Homework Equations N/A The Attempt at a Solution...

Hey all, Joined a Material Science program and I have a course on Multivariate Stats. I have pretty minimal background on the topic so I thought I would pop back here asking for recommendations for a good start on the topic. Here's the course description -

I was watching a lecture on youtube about linear regression and there's a section where it had the statement below (written in purple). Does multiplying by sigma rotate the distribution to make it look like x - N(mew, sigma^2)? Mew in this case is 0 so it doesn't shift the distribution.

I have proved (8.1). However I am trying to prove that ##\bar{X},X_i-\bar{X},i=1,...,n## has a joint distribution that is multivariate normal. I am trying to prove it by looking at the moment generating function: ##E(e^{t(X_i-\bar{X})}=E(e^{tX_i})E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})## I am...

So say we want to see if child birth order causes down syndrome. Say we get that child birth order is statisitcally associated down syndrome. We know that this association could be explained by a third variable: age. Women that give birth to say their third child, is of course going to be...

So ##\vec Y##~MVN(X##\vec\beta##, ##\sigma^2##I) and ##\hat {\vec Y}##~MVN(X##\vec\beta##, ##\sigma^2##H) and I want to show ##\hat{\vec e}##~MVN(##\vec 0##, ##\sigma^2##(I-H)) Where ##\hat{\vec e}## is the vector of observed residuals( ##\vec e=\vec Y- \hat{\vec Y}=(I-H) \vec Y ##). And...

Homework Statement Hi, I am trying to follow my book's hint that to find the normalization factor one should "Diagnoalize ##\Sigma^{-1}## to get ##n## Gaussian which will have variance given by the eigenvalues of ##\Sigma## . Then integrate gives ##\sqrt{2\pi}\Lambda_i##, then use that the...

Are there any good examples of how group theory can be applied to solve multivariate algebraic equations? The type of equations I have in mind are those that set a "multilinear" polynomial (e.g. ## xyz + 3xy + z##) equal to a monomial (e.g. ##x^3##). However, I'd like to hear about any sort...

Homework Statement Show that ##f: \mathbb{Z}^{+} \times \mathbb{Z}^{+} \longrightarrow \mathbb{Z}^{+} ##where ##\displaystyle f(m,n) = \frac{(m+n-2)(m+n-1)}{2}+m## is bijective Homework EquationsThe Attempt at a Solution First, we show that ##f(a,b) = f(c,d) \implies a=c \land b=d##. ##f(a,b)...

Homework Statement A probability density function, ##p\left(x\right)=\int _{-a}^a\left(f\left(x,\left(a-y\right)\right)-f\left(y,ax\right)\right)dy##, can be formed from the bivariate function ##f\left(x,y\right)=y\sqrt{\left(x-2\right)^2-y}## over the domain ##[0,b]## - where ##b## is the...

hello guys, i have a question and looking for an answer quickly... question; prove that, Z is a vector.thank you.

I've taken the single and multivariate calculus classes at my school, (college class offered in my high school for accelerated students). I'm currently a junior, but in the summer of my senior year, I plan to read a book on proofs, brush up on math glossed over by the american education system...

Homework Statement A fair coin has a ##1## painted upon one side and a ##2## painted upon the other side. The coin is tossed ##3## times. Write down a sample space for this experiment. Let ##X_1## be the sum of the numbers obtained on the first ##2## tosses and ##X_2## be the sum of the numbers...

Homework Statement The random variable ##(x,y)## has density ##f(x,y) = ce^{-(ax+by)}## for ##0\leq y\leq x\leq 1##, with given constants ##a > 0##, ##b > 0##. 1. Compute the constant ##c##. 2. Find the conditional probability density ##f_y(y|x)##. 3. Compute the regression curve of ##Y## on...

Homework Statement https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202016-02-02%20kl.%2007.35.26.png I want to find variance matrix and expected variance vector of Y=(Y1,Y2). Y1 and Y2 are independant. Γ is the gamma function and ϒ is a known parameter. λ1>0 λ2>0 and ϒ>0...

Let us assume the validity of Ampère's circuital law\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}where ##\mathbf{B}## is the magnetic field, ##\gamma## a closed path linking the current of intensity ##I_{\text{linked}}##. All the derivations of the Biot-Savart law for a...

As a undergrad future scientist, in my free time I like to do some programming to get some practice as my education programme doesn't include it, but it is a useful skill to have. I am programming some type of roguelike game, so I can get away from graphics and focus on procedural generation...

Hello everyone, I am currently considering a set of random variables, \vec{x} = [x_1,x_2,...x_N] which are know to follow a multivariate normal distribution, P(\vec{x}) \propto \mathrm{exp}(-\frac{1}{2}(\vec{x}-\vec{\mu})^\mathrm{T}\Sigma^{-1}(\vec{x}-\vec{\mu})) The covariance matrix Σ and...

Homework Statement This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of...

Suppose I have a programme. In there I have the creation of histograms: signal: histo_s_Pt histo_s_v1 histo_s_v2 background: histo_b_Pt, histo_b_v1 histo_b_v2 iteration to fill signal/bckg from events j. So at this point I'm having the 6 above histograms filled. Now if I wanted to compare...

I am trying to understand multivariate linear regression. I have a list of time that it took running processes based on several params, like % of cpu usage, and data read. Eg, I have a process that took 50 seconds to run, with a cpu usage of 70%, and the process read 10bytes of data. I have...

Dear all, I have more than 1500 matrices that are the occurency frequency of a bivariate dataset, something like the following: [0.1, 0.3,0.1;0.4,0.05,0.05]. I need to extract the few of them that are really different from the others. Instead of visually analyze each of them and trying to...

1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...

Homework Statement differentiate the function F(x,y) = f( g(x)k(y) ; g(x)+h(y) ) Homework Equations Standard rules for partial differentiation The Attempt at a Solution The Jacobian will have two columns because of the variables x and y. But what then? f is a multivariate function inside...

Homework Statement max/min of f(x,y) = x + y constraint xy = 16 The Attempt at a Solution With lagrange multipliers I did ## \nabla f = (1,1) ## ## \nabla g = (y,x) ## ## \nabla f = \lambda \nabla g ## ## 1 = \lambda y ## ## 1 = \lambda x ## Since y=0, x=0 aren't a part of xy = 16 I can isolate...

Please suggest, book on 'Partial Least Square.'

"In multivariate probability distribution higher-order cumulants contain information of decreasing significance, unlike higher-order moments".

Homework Statement Trying to figure our how to solve the following: \frac{dW}{dσ} where W(σ) = 2π\int_0^∞y(H(x,σ))x,dx Homework Equations both y and H(x,y) are continuous functions from 0 to Infinity The Attempt at a Solution Tried using the leibniz rule but it's not really...

What is the maximum number of roots of a multivariate polynomial over a field? Is there a multivariate version of the fundamental theorem of algebra?

Hi all, I have no experience with multivariate statistics but I'm trying to figure out a type of problem that seems like it should be easy, but I don't know where to begin. I ran into the problem in my research, and I don't have a strong statistics background. I'll try to explain it with an...

Hey all! I have just finished a course in Calc II and have started a course in multivariate calculus. Is this right course I should have taken? Also, what does a typical order for math look like? I am in my last semester of high school; so far my career in math looks like this: MiddleSchool...

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...