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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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Expectation of multivariate gaussian w.r.t. other multivariate gaussian

I want to calculate the Kullbach Leibler Divergence of two multivariate Gaussians as in KL divergence between two multivariate Gaussians. At one point one has to solve the following expression (...
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Gaussian-to-gaussian transformations

It is well-known that when a linear transformation is applied to a normally-distributed random variable, the result is itself a normally-distributed random variable. I am interested in the converse ...
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Calculating the CDF value of a random variable [closed]

I have the following equation: It seems that there is a random variable inside the CDF brackets of a normal distribution in the second half of the equation ($z_1$). It seems that we can use ...
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Using $r_Q$ statistic to check multivariate normality

Suppose in R, I have a data set data(iris) and I want to check for multivariate normality of the first four columns ("Sepal.Length" "Sepal.Width" "Petal.Length"...
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Is Cov(X,Y|Z).x always positive? (with X,Y,Z, normal random vectors and x>0)

Let x be a vector of positive values, we know that for multivariate normal distributions of X, Y and Z, $Cov(X,Y|Z)x=(\Sigma_{XZ}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YZ})x$ does not depend on the given ...
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two-sample $T^2$ testing with $n=1$

Assuming I have two populations $X\sim\mathcal{N}(\mu_X,\Sigma)$ and $Y\sim\mathcal{N}(\mu_Y,\Sigma)$ of sizes $n_X,n_Y$, the hypotheses for equivalence of means (each of length $p$, and assuming ...
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Estimating means and covariance matrix of a multivariate normal distribution

Estimating the covariance matrix of a multivariate normal distribution needs to ensure positive definite constraint. I found there is an Appendix of a paper that describes using proximal gradient ...
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Normal distributed random variables with constraint?

Consider $n$ random variables $X_i$ with $i=1,2,...,n$, each drawing values from identical normal distributions with mean $\mu=0$ and standard deviation $\sigma=const.$ so that expectation values are $...
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Calculating P-value for multivariate normal distributions?

I'm given a set of 500 ($\sigma_i, \mu_i$) that define a 500-dimensional multivariate normal distribution, where each dimension is effectively independent from all the others. Given a 500-...
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Prediction Interval on a Product

As a (perhaps) less offensive twist on this question, suppose that $z_i$ are independent $p$-variate standard normals: $z_i \sim \mathcal{N}\left(0, I\right).$ Let $a$ be an unknown $p$-vector. ...
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Transformation BX+mu from Independent normal to multivariate normal

I was reading https://faculty.math.illinois.edu/~r-ash/Stat/StatLec21-25.pdf (21.1 to 21.5) Where the material suggests getting Multivariate Gaussian $\vec Y \in R^n$ from Independent Gaussians $\...
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Probability of Ranking

Suppose that I have 3 normal distributions (I wish to extend this to K), such that $p_k\sim\mathcal{N}(\mu_k,\sigma_k^2)$. Assume that $(\mu_k,\sigma_k^2)$ are known. How would you go about ...
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Distribution of the average of multivariate normals?

I have seen that the sum of $n$ iid multivariate normal vectors (mean $\mu$ and variance $\Sigma$), $X_1+\dots+X_n$, is distributed as a normal with mean $n\mu$ and variance $n\Sigma$. Is the ...
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How to “honestly” calculate likelihood of 2D normal with small sample size? [closed]

I estimate covariances from data and want to calculate likelihood. For 1D case I know - if the sample size is $<40$, I use Student's t-distribution to calculate likelihood of the data since my ...
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Valid covariance matrix?

I am trying to replicate some results from this well-cited paper evaluating various methods to determine the rank of a matrix. They give several covariance matrices, and then sample from a Gaussian ...
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Location of “x standard deviation(s)” for multivariate normals

In 1D with scalar parameters $(\mu,\sigma^2)$, it is common to represent normally distributed data with error bars spanning $[\mu-\sigma, \mu+\sigma]$. In n-D with parameters $(\mu, \Sigma)$, where $\...
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Distribution of Linear Transformations of Gaussian Marginals

Suppose I have random variables $X = (x_{1},x_{2},x_{3}) \sim \mathcal{N}(0, \Sigma_{x})$. Now let $\tilde{X} = (x_{1},x_{2}) \sim \mathcal{N}(0, \Sigma_{x1x2})$ and suppose I compute a new random ...
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I want to create a random sample of length n from a normal multivariate distribution

This may sound like a stupid question, but I have a problem in understanding this question, especially this part: "generate a random sample of length n from a normal multivariate" This is what I ...
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GMM model of the joint distribution from multivariate marginals

I have two multivariate Gaussian variables $X \sim \mathcal{N}(\boldsymbol {\mu}_X \in \mathcal R^d,\boldsymbol {\Sigma}_X \in \mathcal R^{d \times d})$ and $Y \sim \mathcal{N}(\boldsymbol {\mu}_Y \...
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Maximum entropy distribution on the hypercube

Given the first two moments, the maximum entropy distribution over $\mathbb{R}$ is known the be the normal distribution. What is the analogue for a distribution over $[0,1]$ given either only the ...
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44 views

Proof of Pearson-Aitken selection formula

I am trying to understand the proof of the Pearson-Aitken selection formula, widely used in statistical genetics. A proof that the formula is general is given by Aitken (1936). However, I failed to ...
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Multivariate Normal : expectation of X given Y is doubly-truncated

Let $(X, Y)$ be distributed as a multivariate normal with parameters $$ \mu = \begin{bmatrix} \mu_X \\ \mu_Y \end{bmatrix} \qquad \Sigma = \begin{bmatrix} \sigma_X^2 & \sigma_{XY} \\ \sigma_{XY} &...
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How to justify that $(Y_1,Y_2)$ is not bivariate normal without finding its exact distribution?

Suppose $X_1$ and $X_2$ are independent $N(0,1)$ variables. Define $$Y_1=X_1\,\text{sign}(X_2)\quad,\quad Y_2=X_2\,\text{sign}(X_1)$$ I have to show that $(Y_1,Y_2)$ is not bivariate normal ...
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Math questions in Kalman filter equation derivation

I am interested in data analysis. While my working data (actually it's shopping mall's daily sale) is accumlating, I wish to find some statistical laws underlying business phenomena. I left school for ...
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If $y \sim \alpha x + \beta + N(0, \sigma ^{2})$ and $x\sim N(\mu, \sigma _{x}^{2})$, what is $P(x,y)$?

If y is linearly dependent on x such that the result of performing a linear regression gives you $y=\alpha x + \beta + \eta$, where the noise is normally distributed with zero mean and some prediction ...
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Conditional distribution between jointly normal and univariate normal random variables

Let $X = (X_1, X_2, X_3)^T\sim N_3(\mu, \Sigma)$, where $$\mu = \begin{pmatrix} 1 \\ 1 \\ 1 \\ \end{pmatrix}, \Sigma = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 &...
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Understanding this expression of the multivariate t-distribution

I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows $$ G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\ Z = N (...
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Is Wikipedia wrong about the Multinormal Distribution PDF

I was on the Wikipedia page of Multivariate Normale Distribution and I was wondering if the PDF of the Multivariate Normale Distribution is right. Shouldn't be: $(2\pi)^{(-k/2)}|\Sigma|^{-(1/2)}$ ...
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219 views

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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64 views

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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21 views

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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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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1answer
39 views

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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2answers
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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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185 views

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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32 views

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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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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260 views

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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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_{...