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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220 views

Conjugate prior for multivariate with known mean and covariance known to a constant

I have a linear trend model (evolving mean and slope) embedded in a larger state space time series model that I would like to constrain to be a spline. With that assumption, the mean and trend ...
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55 views

Multivariate distribution for products of random variables

Suppose I have an $n$-dimensional complex, zero mean normal distribution with covariance matrix $\Sigma$, which is not diagonal. Denoting each of the random variables as $x_1, \dots ,x_n$ I would ...
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123 views

Sampling distribution of variances of multivariate normal RVs

Is there an analytical expression for the distribution of variances of MVN RVs? I mean if $X=[x_1, \dots, x_D]\sim \mathcal{N}(0, \Sigma)$ where $\Sigma$ is a $D$-dimensional covariance matrix, is ...
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60 views

Evaluate the multivariate normal using variance matrices $\boldsymbol{\Lambda}+\alpha_{i}\mathbf{a}\mathbf{a}^{T}+\beta_{j}\mathbf{b}\mathbf{b}^{T}$

I need to calculate a huge amount of inverses and determinants to evaluate the pdf of the multivariate Gaussian. Specifically I need to compute the inverses and determinants of the following ...
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1answer
6k views

Marginal, joint, and conditional distributions of a multivariate normal

Let $Y$ ~ $MVN_3(\mu, \Sigma)$ where $\mu = (5,6,7)$ and $\Sigma = \begin{bmatrix}2 & 0 & 1\\0 & 3 & 2\\1&2&4\end{bmatrix}$ Find (a) The marginal distribution of $Y_1$ (b) The ...
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1answer
2k views

Proof for linear combination of multivariate normal X?

Can anyone link to a proof for both parts of the statement below? I assume this question has been asked before but I wasn't quite sure what to search and couldn't find anything. If $X$ is ...
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87 views

Find an unbiased estimator of $\Sigma^{-1}$

Suppose $ X_1,\dots, X_n$ be a random sample from $N_p(\mu, \Sigma), \Sigma > 0$. Find an unbiased estimator of $\Sigma^{-1}$. I know the unbiased estimator of $\Sigma$ is $\dfrac{1}{n-1} \sum_{j=...
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72 views

Probability that the entries of a sample covariance matrix of a m.v. Gaussian exceeds a threshold

Let $\Sigma$ be a covariance matrix of $n$ data points in $\mathbb{R}^p$. So $\Sigma$ is $p\times p$. Suppose that the $n$ points are drawn from the distribution $\mathcal{N}(\vec{0},\Sigma_{\text{...
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897 views

Clarification on LDA and the multivariate Gaussian

From my understanding, to calculate the posterior probability of a sample $x$ belonging to a class $k$ using Linear Discriminant Analysis you would first calculate the eigenvector matrix $W$ required ...
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2answers
1k views

Book about the normal distribution and multivariate normal distribution

I have a problem concerning the normal distribution and multivariate normal distribution. Is there an entire textbook that focuses on these distributions?
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1answer
528 views

Quadratic form of a bivariate normal

This is a homework problem. Let $(X,Y)\sim N(\mu_1,\mu_2,\sigma^2_1,\sigma^2_2,\rho)$. Show that if $\sigma_1,\sigma_2 >0,|\rho|<1$, then $$ \dfrac{1}{1-\rho^2}\left\{\dfrac{(X-\mu_1)^2}{\sigma^...
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How to determine quantiles (isolines?) of a multivariate normal distribution

I'm interested in how one can calculate a quantile of a multivariate distribution. In the figures, I have drawn the 5% and 95% quantiles of a given univariate normal distribution (left). For the right ...
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5answers
4k views

Generate normally distributed random numbers with non positive-definite covariance matrix

I estimated the sample covariance matrix $C$ of a sample and get a symmetric matrix. With $C$, I would like to create $n$-variate normal distributed r.n. but therefore I need the Cholesky ...
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1answer
258 views

Conditional distribution of quadratic forms

Given that $Y$ follows multivariate normal distribution ,i.e, $N_n (0, \sigma^2 I_n)$, we want to find the distribution of $Y'Y$ given that $a'Y=0$ where $a$ is a non zero constant vector. I know ...
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227 views

Re-parametrizing Gaussian distribution?

Let there be observed data points $X = {X_1.. X_n .. X_N}$, where each $X_n \in R^D$. Lets assume these are distributed as a Gaussian $X \cong \mathcal N(\mu,\Sigma)$. Let us also assume that the mean ...
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1answer
20k views

Generating values from a multivariate Gaussian distribution

I am currently trying to simulate values of a $N$-dimensional random variable $X$ that has a multivariate normal distribution with mean vector $\mu = (\mu_1,...,\mu_N)^T$ and covariance matrix $S$. I ...

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