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Questions tagged [multivariate-normal]

The multivariate normal distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. (Also called, multivariate Gaussian)

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Mutual Independence in a Multivariate Normal with Identity Covariance

Consider a random vector $X$ which follows a multivariate nomal with zero means and Identity Covariance. $X\sim \mathcal{N}_n(\mathbf 0, \mathbf I)$ We can say that the individual variables $X_1, ...
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multivariate normal distribution with mean vector 0 and covariance matrix Σ

I am newby in statistics and I have huge data with "p" variables and "n" samples. My data is a two dimensional matrix with "n" columns (each column is a sample) and "p" rows (each row is a variable). ...
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In multivariate error function: why $\frac{1}{2n}$ in $E(w)=\frac{1}{2n} \|Xw -t \|_2^2$?

In multivariate error function: why $\frac{1}{2n}$ in $E(w)=\frac{1}{2n} \|Xw -t \|_2^2$? What does it do? What is it related to?
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Unbiased estimator for Theta of a Normal Distribution

If $X_1,\ldots,X_n\sim \operatorname{iid} \operatorname N(\theta, \sigma^2)$, then verify that $\bar{X}_n$ is unbiased estimator for $\theta$ and that Cramer Rao bound is met? I am facing difficulty ...
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How to swap variables in a conditional normal distribution?

I assume that I have two normal distributed variables where one depends on the other: $P(A) \sim N(0,\sigma_a)$ $P(B|A) \sim N(q\cdot A, \sigma_b)$ How can I get the reverse $P(A|B)$ assuming that ...
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Finding joint probability distributions from marginal distributions

Question: I was solving test papers where I found this one. My doubt: I know to work with conditional probabilities and Jaccobian Transformation and part A and B can be done applying the above..But ...
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Sample from aggregate portfolio distribution versus individual asset distributions

Suppose I have three assets $x_1,x_2,x_3$ in a portfolio with weights $W=\begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} $, expected returns $R=\begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \end{bmatrix}$, ...
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What will be the distribution of the following quantity?

Suppose $X_1,X_2,\dots,X_n$ are p-dimensional random variables with distribution $N_p(\mu,\Sigma)$. Let $X_{n\times p}$ is the data matrix. $\mathbb{Z}_{n\times p}= (n-1)^{-1/2}\left(I_n - n^{-1}\...
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Long samples from Gaussian Process _prior_

I'm interested in being able to sample a long (N~10^5) sample from a Gaussian process. For a small sample I understand I can quite easily construct an NxN covariance matrix and then choose a random ...
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Fast Approximate Sampling from Multivariate Normal Parameterized by Precision Matrix

I want to efficiently sample $x \sim N(\mu, \Omega)$ where $\Omega$ is a precision matrix (e.g., the inverse of the covariance. The challenge is that the dimension of $x$ is massive (~ 100K to 10M) ...
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Product of two multivariate Gaussian distributions of different dimensions

I am computing the posterior in a multi-output Bayesian regressor. I assume the prior to be a matrix Gaussian distribution. I can write the prior and the likelihood as multivariate normal ...
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Higher order extensions of gaussian distribution?

I anticipate that this question may be predicated on some misconceptions or confusion. Please have patience. Through Bishop's PR&ML book, as well as a little bit of exposure to statistical ...
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Find conditional probability of multivariate normal

Given vector $X \sim N_2((1,2)^T;\begin{bmatrix} 1/2&&3/2 \\ 3/2&&1/2 \end{bmatrix}))$ find conditional probability of $A={(x_1,x_2):x_1^2+x_2^2=3)}$ given that $x_1=2$ So what ...
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Variance of the sum of elements of a Wishart distributed matrix

Looking for the variance of $S=\sigma _{1,3}-\sigma _{1,4}-\sigma _{2,3}+\sigma _{2,4}$, where $\sigma_{i,j}$ are Wishart-distributed elements of the random matrix $$\Sigma =\left( \begin{array}{cccc}...
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Definition of independence of two random vectors and how to show it in the jointly normal case

(1) What is the definition of independence between two random vectors $\mathbf X$ and $\mathbf Y$? I think it's more than just pairwise independence between all the elements $X_i$ and $Y_j$. (2) The ...
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Distribution of the dot product of a multivariate gaussian random variable and a fixed vector

If $a$ is a multivariate normal random variable, and $x$ is a plain old vector (of the same shape as $a$), then the inner product $x \cdot a$ is a random variable. This post on math exchange suggests ...
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Can we use covariance matrix to examine feature collinearity?

Consider using Multi-variate Gaussian to approximate $X = [X_1, X_2, ..., X_n]$ and $X_i = [x_{i1}, x_{i2}, x_{i3}, ..., x_{im}]$, so we have n data points and each data point has m features. Multi-...
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D-dimensional Gaussian posterior distribution

A $D$-dimensional Gaussian random variable $𝑥$ with distribution $N(x| μ, Σ)$ in which the covariance $Σ$ is known and for which we wish to infer the mean $μ$ from a set of observations $X = {𝑥_1, . ...
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Is Drawing Each Vector Component from Univariate Normal the same as Drawing from Uncorrelated Multivariate Normal?

So I'm thinking about drawing a vector $x \in \mathbb{R}^k$ and I'm trying to figure out if drawing each component $x_i$ from a univariate normal $\mathcal{N}(0, \Sigma_{ii})$ with $\Sigma_{ii} \in \...
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Chernoff bound for multivariate normal distribution

I read in Introduction to Statistical Pattern Recognition about different bounds for Bayes classification errors. It asked to prove that for two multivariate normal distributions, a Chernoff bound ...
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When are correlated Normal random variables multivariate Normal?

I know that there are many example of correlated normal random variables which are not jointly (multivariate) normal. However, are there conditions which state when correlated normal random variables ...
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Joint probability of multivariate normal distributions with missing dimensions

Suppose I conduct two experiments, each measuring a subset of possible parameters. From experiment #1 I measure two parameters and estimate the multivariate normal distribution $$ \mathbf{X}_1=\left [...
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Equality of two multivariate normal CDF's

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
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Multivariate Normality ANOVA test

ANOVA requires multivariate normality as a key assumption. I understand that this assumption can be tested in MANOVA using the Mardia test, Royston test, and Henze-Zirkler test. This can be ...
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Variance of a Hermitian quadratic form of complex normal vector

Suppose I have a random vector $\mathbf{v} = (v_0, v_1, \ldots, v_n)$ where each entry is a complex number, so $\mathbf{v} \in \mathbb{C}^n$ and $\mathbf{v} \sim \mathcal{CN}_n(\pmb{\mu}, \pmb{\Sigma}$...
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fraction below range for normally distributed population

We have months of continuous glucose sensor data from hundreds of subjects, sampled every 5 minutes. For each subject we calculate $\mu$ and $\sigma$ (normal distribution). For the entire subject ...
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Proving independence between linearity and quadratic forms using independence of linear and quadratic form

I'm reading through these lecture notes online http://www.pitt.edu/~wahed/teaching/2083/fall09/Lecture309.pdf And on page 103 he notes the following theorem If $X \sim N(\mu, \Sigma)$ and $A^T = A$...
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Expectation of expressions involving sample covariance matrix and inverse of covariance matrix

Let $y_{ij}$, $i=1,2,\cdots,n_j$ be a random sample from $N_p(\mu_j,\Sigma_j)$, $j=1,2$. Let $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$ $$S_j=\frac{1}{n_j-1}\sum_{...
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Should updating one data point at a time or all change the posterior of a normal-inverse-gamma?

I have implemented the normal inverse gamma distribution per section 3 of https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture5.pdf in some code. However, I've noticed ...
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Distribution of the sample variance of values from a multivariate normal distribution

What is the sampling distribution of the variance of a collection of variables that follow a multivariate normal distribution? Specifically, assume that the $n-$dimensional vector $\boldsymbol{x} \sim ...
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Formulation for multiple regression, but with the bias term taken out and treated separately [duplicate]

Does anyone have a reference to an explicit formulation of multiple regression, but in which the bias term is taken out and treated separately? I would especially be interested if either ridge ...
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Prove identity involving two multivariate normal distributions

Let $y_{ij}$, $i=1,2,\cdots,n_j$ be a random sample from $N_p(\mu_j,\Sigma_j)$, $j=1,2$. Let $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$ $$S_j=\frac{1}{n_j-1}\sum_{...
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Fourth order moments of the sum of multivariate normal distribution

Suppose $X\sim \mathcal{N}_d(\boldsymbol{0},\boldsymbol{\Sigma})$ follows a $d$-dimensional multivariate normal distribution. Let $\text{A}_i$ and $\text{A}_j$ be two arbitrary symmetric $d\times d$ ...
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General restriction for covariance matrix in multivariate normal distribution

Suppose we look at the following model $$ \vec y_i=\vec\mu_i + \vec\epsilon_i, \qquad \vec\epsilon_i\sim N(\vec0, \Sigma) $$ where $\vec y_i$s is observed, $\vec\mu_i$s are known, and $\vec\...
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Variance of the product of correlated normal variables

Suppose that $X$ and $Y$ are scalar random variables that are jointly normally distributed: $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim N \left( \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix}, \begin{...
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Does assumption of normality of each mixture components implies that each margins is normal

I just would like to understand some information about the joint normality and the margins. I read that the normal joint distribution almost always implies that the univariate margins are all normal. ...
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What is the distribution of two bivariate normal random variables given difference?

Suppose $X_1$ and $X_2$ are bivariate normal with mean $\mu=(\mu_1,\mu_2)$ and covariance $\Sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{12} & \sigma_{22} \...
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How to determine the cut off value of an hyperellipsoid in order to retrieve a single quantile of a multivariate normal distribution?

Introduction My goal is to retrieve the $\alpha$ quantile of a N(0, H) (multivariate normal) random variable $X$ where H is a known d-dimensional positive definite matrix (with $d >3$). In other ...
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How do I find the “elliptical confidence region” from columns of a matrix that follows the Wishart distribution?

The subject is about the sample mean and the sample covariance estimators and their respective confidence regions for the estimated parameters. Suppose that $n$ samples are taken from a $p$-variate ...
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Distribution of quadratic form of multivariate normal with linear term

Suppose that $A$ is a symmetric non-random matrix and $X\sim N(\mu,\Sigma)$ and $b \in R^n$ is a non-random vector. Then what is the distribution of $$X^tAX+b^tX \quad ?$$ The distribution without ...
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How can I create a multivariate random normal distribution from separate normal distributions?

In R I can sample from a multivariate normal distribution as follows: ...
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Fitting a Multivariate Normal Model

Is there a standard method for fitting multivariate normal models where $\mu(\theta)$ and $\Sigma(\theta)$ are nonlinear functions of the model parameters? In my case I have a single vector of ...
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Predictive density in Normal-Wishart model

In various online notes (e.g. here and here) the following fairly standard result is found: if the $d$-dimensional vector $x$ is $x\sim N(\mu, \Lambda^{-1})$ and the prior is $\Lambda\sim W(\Lambda_0, ...
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What does it mean by “The statistic is asymptotically distributed as N(0,1).”

I'm reading a paper Mardia (1974) about multivariate normal tests. There is a line that says "The statistic is asymptotically distributed as N(0,1)." Now, I have calculated this value for one of my ...
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how can I predict a variable using a correlated variable? [duplicate]

I have two variables from a multivariate standard normal distribution, which are highly correlated (say, r = .8). If I observe variable 1 to be 1.5, how can I predict the value (or the possible range ...
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Multivariate Generalisation of Percentile and Likelihood of a Draw

Say I have $n$ values which approximate some distribution $D$. If I have a value $x$ and I would like to know how likely it is that $x$ came from $D$, I could simply determine $x$'s percentile. By ...
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Joint raw moments of multivariate normal

Let $X_1,X_2$ follow a bivariate standard normal distribution with some non-zero correlation coefficient, $\rho\neq 0$. Let the function $f(z) = z^k,\; k=1,...$. By Stein's lemma, we have that $$\...
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Maximum Likelihood Estimators - Multivariate Gaussian

Context The Multivariate Gaussian appears frequently in Machine Learning and the following results are used in many ML books and courses without the derivations. Given data in form of a matrix $\...
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Sufficient conditions for joint normality?

Suppose I have $n$ random variables $X_1,...,X_n$ such that $X_1 \sim \mathcal{N}(0,1)$ and the increments $X_i - X_{i-1} \sim \mathcal{N}(0,s)$ are independent. Are these conditions sufficient to ...
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Hypothesis of the mean test for multivariate normal and known Sigma

In the book Applied Multivariate Statistical Analysis by Härdle and Simar on page 173, they present a Test problem. TEST PROBLEM 1 Suppose that $X_1, \cdots, X_n$ is an i.i.d random sample from $N_p(\...