# 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)

600 questions
Filter by
Sorted by
Tagged with
1 vote
38 views

### 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 ...
30 views

### 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 ...
• 123
39 views

### 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 ...
26 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 ...
• 1
25 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 (...
• 184
106 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 ...
28 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 ...
• 1,263
23 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 ...
22 views

### 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 ...
25 views

### 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 ...
17 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 $k-$...
• 249
1 vote
17 views

### 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 ...
• 249
63 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 ...
• 3,405
81 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 ...
• 3,405
14 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
58 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, $\Sigma$).
• 13
25 views

### Analogous result to Isserlis' theorem for mixed absolute product-moments of multivariate normal distribution

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 ...
• 3,405
1 vote
86 views

• 101