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)
747 questions
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Sampling from mvnorm with n=2 vs sampling twice with n=1
Are these two in theory equivalent in R?
b <- mvrnorm(n = 2, mu = rep(0, m))
b1 <- mvrnorm(n = 1, mu = rep(0, m))
b2 <- mvrnorm(n = 1, mu = rep(0, m))
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Statistical associations between minors of the Ginibre ensemble
(This post is rewritten from an unanswered earlier post of mine at Math SE; I just realized that Cross Validated might be a more appropriate place for it.)
Suppose we have an $n \times n$ matrix $A$ ...
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Correlation of random variables after non-linear transformation [duplicate]
I have the following doubt, probably trivial. Let us suppose to have three normally distributed random variables $ x_1 $, $ x_2 $, and $ x_3 $ inside a multivariate normal probability density function:...
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Multivariate Normal Tail CDF Implementation
Are there any efficient codes that can evaluate the probabilites of a multivariate normal CDF (4-10 variables), in MATLAB or R or Python. Probabiites in the range of 10^-5 to 10^-6
I used MATLAB's ...
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Conjugate prior for normal likelihood when the likelihood covariance is shifted
I have a multivariate normal likelihood of the form
$$p(y| \mu_\alpha, \Sigma_y + \Sigma_\alpha) = N(\mu_\alpha, \Sigma_y + \Sigma_\alpha) $$
where $y$ is observed data, $\Sigma_y$ is known, and $(\...
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Change of Variables in a Double Integral Involving Bivariate Normal Sample and Spectral Decomposition
In my research involving statistical inference for the bivariate normal distribution under a special sampling scheme, I encountered the following integral:
\begin{eqnarray}
I=\int_{-\sqrt{s_{11}}}^{\...
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Why don't we ignore dividing by variance in discriminant functions for the normal density in case where covariance matrix for all classes is sigma*I?
I am studying about Discriminant Functions For The Normal Density (2.6) from the book Pattern Classification (Second Edition) by Duda,Hart and Stork.The book suggest following function for ...
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Covariance matrix construction problem for multivariate normal sampling
I have encountered a problem when trying to simulate from a multivariate normal distribution with a covariance matrix built to encode specific correlations. Most of my parameters are estimated from ...
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Asymmetric Bayes error $\mathcal{N}\left(0,\begin{bmatrix} \sigma_1^2 & 0 \\ 0 & \sigma_2^2 \end{bmatrix}\right)$ vs $\mathcal{N}(0,I)$ classification
Consider the problem of classifying $x \in \mathbb{R}^2$ into one of two classes, $c1$ and $c2$, with known distributions \begin{align} & p(x\mid c1) \sim \mathcal{N}\left(\begin{bmatrix}
0 \\
0
\...
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Test the hypothesis $\langle \mu_1,\mu_2\rangle= \langle \mu_3,\mu_4\rangle$ for four $d$-dimensional groups of normal population
Assume that I have four groups of samples, each from a $d$-dimensional normal population $N(\mu_i,\Sigma_i)$ ($1\le i\le 4$).
All the $\mu_i, \Sigma_i$ are unknown parameters.
I need to test the ...
user398751
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How can we efficiently sample from the truncated multivariate normal distribution? [closed]
As the title says, I would like to know if there any method already implemented in R that efficiently samples from the truncated multivariate normal distribution. More precisely, given $\textbf{X}\sim\...
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How can we generate a nonstandard multivariate normal from a standard multivariate normal? [duplicate]
Is this true: Let $$Z \sim \mathcal N (0, I).$$ Then if $$X = \Sigma^{1/2}Z + \mu$$ then $$X \sim \mathcal N(\mu, \Sigma).$$
That is, we can nonstandardize a multivariate normal in the exact way we do ...
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Is the expression for $\mathbb{E}[\textbf{X}\textbf{X}' \mid \textbf{X} \leq \textbf{C}\textbf{b}]$ correct?
As the title says, I would like to know if the following relation is true:
\begin{align*}
\mathbb{E}[\textbf{X}\textbf{X}' \mid \textbf{X} \leq \textbf{C}\textbf{b}] = \boldsymbol{\Sigma} + \mathbb{E}[...
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Asymptotic variance of MLE of $\boldsymbol \mu$ when $\lVert \boldsymbol \mu \rVert=1$
Suppose $(X_1,Y_1),\ldots,(X_n,Y_n)$ is a random sample from a bivariate normal $N_2(\boldsymbol \mu, I_2)$ distribution where $\lVert \boldsymbol \mu \rVert=1$. Let $\widehat{\boldsymbol\mu}$ be the ...
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Difference between normal and multivariate normal distribution for Y in linear regression?
In linear regression, I'm assuming that when $y$ belongs to a normal distribution it's because there's only one variable, as in it's a simple linear regression, for example $y = \beta_{0} + \beta_1 ...
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