# 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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### Marginal Likelihood Computation for Bayesian Linear Model

Given a simple Bayesian linear model with $N$ observations $y = X\beta + \varepsilon \quad \quad \varepsilon \sim \mathcal{N}(0, \Sigma)$ with known error variance-covariance matrix $\Sigma$ and ...
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### Distribution of exponent of multivariate normal distribution

My slides say that the exponent of a multivariate normal distribution, $(\mathbf{X} - \boldsymbol{\mu})^\text{T} \boldsymbol{\Sigma}^{-1} (\mathbf{X} - \boldsymbol{\mu})$, follows a chi squared ...
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### The principal submatrix of projection matrix with Gaussian design

I've come across a phenomenon from a simulation that I'm very curious about. But I don't know how to start my analysis. So, I am asking for some guidance. Thanks! Denote by $\mathbf{H}$ the principal ...
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### Covariance matrix of multivariate normal when negative values are made zero

Let $x$ be $n$ dimensionally multivariate normally distributed with mean $\mu$ and covariance matrix $\Sigma$. Now let $y$ be random variables defined by \begin{equation} y_i= \begin{cases} 0, ...
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### What is the expected value of $X_i/\|X\|^2$ when $X \sim \mathcal{N}(\mu, \sigma^2I)$

Let $X$ be an N-dimensional normal random vector with non-zero mean $\mu$ and diagonal covariance matrix $\sigma^2I$. I would like to understand if it is possible to derive the expected value of the ...
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### Wrapped normal distribution variance from angle between two multivariate normal distributions

Suppose there are two 2-d multivariate normal distributions , as in the image below: There is no correlation between the x and y components in the distributions, and the variance for each dimension ...
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### How to bound the probability of multivariate gaussian vector norm?

Let's say $v \in \mathbb{R}^n \sim \mathcal{N}(0, \sigma I)$. That is, $v$ is a gaussian random vector, whose entries are distributed $\mathcal{N}(0, \sigma)$ i.i.d. From the book "C. Giraud. ...
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### How can population variance be estimated from a bivariate sample?

Let's assume a bivariate population with a correlation $\rho$ and a common $\sigma$ so that $\Sigma = \sigma^2 \begin{pmatrix}1 & \rho \\ \rho & 1\end{pmatrix}$. I would like to know the ...
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### What is the distribution of the $k^{th}$ highest value of a multivariate normal distribution

Let X be an N-dimensional multivariate normal, $X \sim N(\mu,\Sigma)$ where $\mu$ is Nx1 and $\Sigma$ is NxN. If we take a draw of $X$ from this distribution and then sort $X$ from largest to smallest,...
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