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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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Explanatory variable independent of the response, yet has non-zero beta

My intuition was that if an explanatory variable is independent of the response then in a multiple regression it should have a $\beta$ of zero. Consider however the following very simple example: the ...
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How to characterize the effect of $(\textrm{Diag}(\Sigma^{-1}))^{-1}$ badly approximating $\textrm{Diag}(\Sigma)$

I have an almost singular covariance matrix $\Sigma\in\mathbb{R}^{n\times n}$ that has a few large eigenvalues, followed by many many comparatively very small ev's. If I were to try to approximate ...
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Compute the mean and std of a random variable from its correlation with known random variables?

Context I'm running statistical simulations on IQ distribution among people and I would like to sample the IQ of a child given : The general distribution of IQ following : $$ \mathcal{N}(100, 15.5) ...
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Multivariate normal error with autocorrelation in second dimension

I am trying to forecast my model, but am unsure how to so in terms of error distribution (using mvrnorm). The model itself essentially estimates numbers over time and state (a matrix time x state). ...
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Deriving the conditional distribution of a multivariate normal, for inequalities

This question is slightly related to Deriving the conditional distributions of a multivariate normal distribution. In that question, the following situation was given. If $Y$ follows a multivariate ...
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Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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Integrating with a multivariate Gaussian

I need to figure out the steps to solve the following integral, where $Q(\mathbf{w})$ is a multivariate Gaussian with mean $\overline{\mathbf{w}}$ and covariance $\mathbf{C}$: \begin{align}\int Q(\...
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Estimate parameters of a normal knowing the density function

I have a matrix with 3 columns. The first column contains values of a variable $x_1 \in [-1,1]$. The second column contains values of a variable $x_2 \in [-1,1]$. The third column contains a variable $...
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Distribution of $XY$ when $(X,Y) \sim BVN(0,0,1,1,\rho)$

The question is pretty much in the title, I need to find an approximate distribution of $XY$ when $(X,Y)$ follow a Bivariate Normal Distribution where $X$ and $Y$ are each $N(0,1)$ distributed and $...
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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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1answer
60 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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1answer
68 views

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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1answer
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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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1answer
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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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156 views

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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44 views

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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36 views

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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38 views

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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1answer
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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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24 views

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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1answer
54 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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1answer
164 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 ...