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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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Simplification of bivariate normal $\phi_2(x,y,\rho)$ at $y=y_F$ (i.e. fixing one of the axes)

Suppose we start off with the traditional standard bivariate normal distribution: $$\phi_2(x,y|\rho,\mu_x=0,\mu_y=0,\sigma_x=1,\sigma_y=1)=\frac{1}{2\pi\sqrt{1-\rho^2}}\exp \left(-\frac{x^2-2\rho x y ...
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Using Hotelling's T-statistic to find an elliptic confidence set

The problem: We have samples of sizes ${n_1} = 25,{n_2} = 15,{n_3} = 30$ drawn independently from $N\left( {{\mu _i},{\sigma ^2}} \right),i = 1,2,3$ (normal distributions with same variance). We have $...
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Phase marginal for a multivariate complex Gaussian density

Suppose $z$ is a random variable taking values in $\mathbb{C}^n$ and admitting the complex Gaussian density $p(z;W) \propto \exp{(-\frac{1}{2}z^*Wz)}$, where $W$ is Hermitian. Let $r$ be the vector of ...
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Multivariate Theory: How does the new mean only depend on the conditioned variable?

I'm doing some review of Gaussian Processes and Multivariate Normal Theory. I found a really helpful website here, but I have run into a snag. What does the author mean in the sentence below this ...
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Conditional covariance of multivariate normal tail

Let $X\sim N(\mu,\Sigma)$, $t\in\mathbb{R}$, and $a$ be a non-zero vector of the same dimension as $X$. Define a random vector $Y=X\mathbb{1}(a^\top X\ge t)$, where $\mathbb{1}$ denotes the indicator ...
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Computing posterior based on sum of multivariate normal distribution

Currently I am exploring topics for my undergrad thesis. Although I took a course in Bayesian statistics, I am not yet sure how to proceed in finding the posterior in the following case. I have a d-...
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52 views

Distribution of transformed multivariate log-normal

Let $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$ and $\mathbf{Y} = \text{exp}(\mathbf{X})$. If $Y_i$ is one of the components of $\mathbf{Y}$, what is the distribution of $\frac{\mathbf{Y}}{...
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Validity of confidence interval for $\rho$ when $X\sim N_3(0,\Sigma)$ with $\Sigma_{ij}=(\rho^{|i-j|})$

Suppose $X\sim N_3(0,\Sigma)$, where $\Sigma=\begin{pmatrix}1&\rho&\rho^2\\\rho&1&\rho\\\rho^2&\rho&1\end{pmatrix}$. On the basis of one observation $x=(x_1,x_2,x_3)'$, I ...
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Taking constant out of multivariate normal

For a univariate normal distribution $X \sim N(0, k\sigma^2)$ we can take out the $k$ to get $\sqrt{k}X \sim N(0, \sigma^2)$. In the multivariate normal case is there something similar? If $\textbf{Y} ...
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Asymptotic covariance matrix of the maximum likelihood estimator of the parameters of a multivariate normal distribution

I would like to know how to derive the asymptotic covariance matrix of the maximum likelihood estimator of the two parameters (mean vector and covariance matrix) of a multivariate normal distribution. ...
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proof of independence of X-Y and X+Y when X,Y come from bivariate normal

I have a bivariate normal distribution: $$\begin{pmatrix} X\\Y\end{pmatrix} \sim \mathcal{N}(\mu, \Sigma)$$ where: $$\mu = \begin{pmatrix}\mu_X \\ \mu_Y\end{pmatrix}$$ $$\Sigma = \begin{bmatrix} \...
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How to evaluate multivariate normal integral with conditional upper bounds

Suppose I have independent normally distributed random variables: $x_i \sim N(0,1)$. In my actual application, $i=1,\ldots,30$, but for my example here I'll use $i=1,2,3$. I want to evaluate (either ...
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R2jags: Truncated Multivariable Normal Distribution in JAGS model (three variables)

I’m trying to specify the joint distribution of 3 parameters in JAGS using a truncated multivariate normal (one of the 3 parameters is truncated at zero, the others are untruncated). I’ve ...
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Derive multivariate from bivariate normal distribution

Could anyone help me on the following. Let $K$ and $M$ be integers so that $K\geq3$ and $2\leq M < K$. Let $\boldsymbol{X}=(X_1, ..., X_K)^T$ be a random vector, $\boldsymbol{\mu}$ be a $K\times 1$...
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Fitting a multivariate Gaussian with extremely sparse samples

We have a multi-variate Gaussian distribution. For instance with 3 variables. The correlations between the variables are important! We are fitting it to data, however, the samples are such that each ...
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Observed information matrix with multivariate normal distribution

$$ \DeclareMathOperator\tr{tr} \DeclareMathOperator\vecOP{vec} \newcommand\di{\mathrm{d}} \newcommand\D{\mathrm{D}} \newcommand\Hess{\mathrm{H}} $$ I do not have much experience with matrix ...
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Help with normalising data that has A LOT of 0s [duplicate]

I recently am analysing my results (behavioural, observation-based data), and I realised that my data are non-normal. No problem, this happens in behavioural data a lot, and I thought I just needed to ...
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Does standard distance follow the 68-95-99.7 rule?

I'm wanting to do a simple standard distance demonstration for my students in R, but I've come across a conundrum. When I simulate the creation of 10,000 points in a spatial normal distribution, ...
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Finding the marginal distribution

If $p(y) =N(0,\Gamma) $ and if $p(x|y) =N(\Lambda y, \psi I) $where $I$ denotes the identity matrix. Also, we have the conditions: $\Gamma_{ij} =0$ for $i \neq j$ (diagonal) $\Lambda \Lambda^T = I$ ...
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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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KL divergence between sample and true (multivariate normal) distribution

I was wondering, whether there is a possible interpretation of the KL-Divergence between sample and true distribution in terms of probabilities. E.g. given $P=\mathcal{N}\left(\mu,\Sigma\right)$ and $...
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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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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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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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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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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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1answer
118 views

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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1answer
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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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1answer
134 views

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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311 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 ...