# Questions tagged [multivariate-normal-distribution]

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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### Probabilistic Machine Learning: Product of gaussian pdfs of samples is equal to gaussian pdf of sample mean

I'm currently reading the book Probabilistic Machine Learning: An Introduction by Kevin P. Murphy and I'm stuck on the derivation of a formula in section 3.3.4 (Example: inferring an unknown vector) ...
1 vote
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### Joint normality of a vector derived from joint normal vectors

Suppose that we have two random vectors following joint normal distributions: $$X=[x_1,x_2]'\sim N(0,\Sigma_X)\quad \textrm{and}\quad Y=[y_1,y_2,y_3]'\sim N(0,\Sigma_Y).$$ In this setup, I am ...
1 vote
40 views

### Joint normality of RV made up of jointly normal RVs

I'm reading a paper where (simplifying): $Y_{i} = \beta_\theta \theta_i + \beta_x X_i + \epsilon_i$, $\epsilon_i \sim N(0, \sigma_\epsilon^2) \perp (\theta_i, X_i)$, and $\beta_\theta$ and $\beta_x$ ...
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### Joint Normality of a Random Vector with elements from a Joint Normal Distribution

Suppose that $X=[X_1,X_2,X_3,X_4]\sim N(0, \Sigma)$ (i.e. $X$ follows a joint normal distribution). Define $Y=X_1+X_2$ and $Z=X_3+X_4$. Here, as far as I know, each of $Y$ and $Z$ is normally ...
26 views

### The distribution function for for the sum of variables which follow a multivariate normal? [duplicate]

Suppose we have two random variables that follow a multivariate normal distribution, $[x,y]\sim MVN([a,b], \begin{bmatrix} \rho_1 & \rho_3 \\ \rho_3 & \rho_2 \end{bmatrix})$ Then what is the ...
39 views

### Generating data in the desired correlation structure from the multivariate normal distribution in R

I want to derive 200 variables from the multivariate normal distribution. I will divide these 200 variables into 3 blocks. The 1st block will consist of 40 variables and will be low-moderately ...
47 views

### What's the conditional variance of the chain X -> Y -> Z?

If I have a cascade of 3 random variables, represented as a Bayesian Graph: $X\rightarrow Y \rightarrow Z$, is there a simple formula for $\sigma_{X|Z}$? Further, assume all the variables are normal, ...
1 vote
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### How to derive solution to loss function of GLasso for precision matrix

I am trying to find the parameter $\hat\omega = min_{\omega}\Big(-log|\omega| + tr(S\omega) + \sum_{i,j}\lambda|\omega_{ij}|\Big)$ This is to regularize the precision matrix $\omega$ for the GLasso. ...
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### How can I compute rectangular confidence regions for parameters using R?

Simultaneous confidence regions for multivariate parameters (say, a confidence region for multivariate mean, or for regression parameters) usually find an elliptical region when the parameters' ...
34 views

### Correlation of multivariate distributions without "slope"

Wikipedia has this image showing different correlations: It says: The correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (...
1 vote
38 views

### Non-positive definite matrix problem for desired correlation structure

I want to derive a correlation matrix such that block1 is 0.1 within itself, block2 is 0.1 within itself and 0.7 with block1, and the remaining variables are 0.01 within itself and with other blocks ...
1 vote
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### Is it possible to derive the joint probability distribution of squared OLS residuals under the classical linear regression assumptions?

Consider the linear regression model, $$\boldsymbol{y}=\boldsymbol{X\beta}+\boldsymbol{\epsilon},$$ where $\boldsymbol{y}$ is an $n$-vector of responses, $\boldsymbol{X}$ is an $n\times p$ matrix of ...
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### Generating multivariate random variable with normal and exponential marginals

I have a collection of data points of the form $[U, V, X, Y]$, where $U$ ~ $N(\mu_1, \sigma_1)$; $V$ ~ $N(\mu_2, \sigma_2)$; $X$ ~ $exp(\lambda_1)$; and $Y$ ~ $exp(\lambda_2)$, and I am looking to ...
1 vote
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### Statistical analysis (comparison) of time course experiments

I have data sets that represent multiple measurements over time. The data sets come from biological experiments in which we measure something in N individual cells, with around 250 measurements over a ...
29 views

### Sum of exponential of MVN dimensions

Consider drawing N random variables $x_{i}$ for $1\leq i \leq N$ from a multivariate normal with $\mu = (\mu_1, .. \mu_i, .. \mu_N)$ and $\Sigma_{N \times N} = [\sigma_{ij}]$ (equivalently, N ...
29 views

### Correlation matrix from pairwise correlations with specified structure

I need to simulate multivariate normal samples with a pre-specified correlation structure. The structure is such that the bigger the (GPS) distance between two points, the smaller the correlation (...
114 views

### Density of Multivariate Normal Distribution (MVN): Dimension = 1 or k?

I have a very naive question about the density function of the MVN distribution. According to the wiki page, the density function formula sometimes has a constant k ...
51 views

### How to split the data into multiple (normal) distributions or clusters?

I am trying to train a machine learning model to predict yield on fields, based on multiple parameters, such as elevation, humidity, and nitrogen content. Observing the historical harvest data, I ...
29 views

### Multivariate Gaussian probability mass inside a sphere

Assume I have some d-dimensional multivariate gaussian $X\sim\mathcal{N}\left(\mu,\Sigma\right)$ and some sphere $C=\left\{ x:\left\Vert x-z\right\Vert_2\le r\right\}\subseteq\mathbb{R}^{d}$. I was ...
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### Linear transformation between two multivariate normal distribution

Suppose I have two multivariate normal distribution $N_1$, $N_2$. If I know the mean and cov of $N_1$, $N_2$：$\mu_1$, $\Sigma_1$, $\mu_2$, $\Sigma_2$. Can I find a linear transformation which makes ...
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### How to generate a Multivariate Normal Distribution out of means and a correlation coefficient in R?

The means are 109 and 115, the correlation coefficient is 0.797153. I know about the mvrnorm() function, but it requires a "a positive-definite symmetric matrix specifying the covariance matrix ...
18 views

I have got a dataset of $N$ observations with $k$ predictors $x_1, ..., x_k$ and a vector $y$ of binary responses, where $y \in \{0,1\}$. I assume that given a class my data comes from a $k-$...
1 vote
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### Overlap coefficient for two multidimensional normal distributions

For two PDFs $f_1(x)$ and $f_2(x)$ the overlap coefficient (OVL) measures the similarity between two distributions through the overlapping area of their distribution functions and is given by the ...
68 views

### Joint distribution of sample correlations of variables taken from a multivariate normal distribution

Let us assume multivariate normal vector $(X_1, \cdots, X_n)$ with mean vector $\mu$ and variance-covariance matrix $\Sigma$. A sample correlation will not exactly equal its population parameter, but ...
85 views

### Can we rule out $\frac{1}{2^n} + \frac{1}{2 \pi (n-1)} \left( \sum_{\substack{i,j \in \{ 1, \cdots, n \} \\ i < j}} \sin^{-1} (\rho_{i,j}) \right)$?

I am curious about orthant probabilities for the multivariate normal distribution for any finite dimension $n$. While Wikipedia currently doesn't seem mention these quantities the Wolfram Mathworld ...
18 views

### calculate correlation coefficient with singular values of covariance matrix

Given a normal distribution where the covariance matrix $\Sigma$ has known singular values $s_1$, $s_2$, ... $s_m$, what are the Pearson's correlation coefficient values?
1 vote
60 views

### Expected Value of $x_1 \exp(a_1 x_1 + a_2 x_2 ... a_n x_n)$ when X is multivariate $N(0, \Sigma)$ [closed]

Which is the $E(x_1 \exp(a_1 x_1 + a_2 x_2 + \dotsm + a_n x_n)$ when X is an n-random vector distributed multivariate normal (0, $\Sigma$).
Suppose that $(X_1, \cdots, X_n)$ have a joint normal distribution. If $n = 2m + 1$, then $\mathbb{E} \left[ \prod_{j=1}^n X_j \right] = 0$. This can be argued from the symmetry of the multivariate ...