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Questions tagged [multivariate-distribution]

Probability distribution over vectors (as opposed to univariate distributions that are over numbers).

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Sufficient conditions for multivariate MGF to be finite in neighborhood of zero

In the one-dimensional case, we have the following fact: (see Existence of the moment generating function and variance for proof) Proposition: The mgf $m(t)$ is finite in an open interval $(t_n,t_p)$ ...
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Multivariate Random Variables – Are the calculations correct? [closed]

Upfront: Sorry for not taking the time to translate it into LaTeX code. I'll do that when I am done with my exams... Simply wanted to ask whether what I am doing is correct, as I am sort of unsure ...
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Estimating Pearson correlation for multivariate t distribution using Kendall rank correlation

I had a task to generate a certain amount of samples for $(X_{1},X_{2})$ with a bivariate $t$ distribution $t_{2}(\nu,\mu,\Sigma)$ and then estimate Pearson's correlation coefficient $\rho$ by using ...
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14 views

Projection of multivariate distribution to lower dimensional subspace

Say that $X \in \mathbb{R}^n$ is a vector of $n$ r.v.'s with pdf $p(x_1,\ldots,x_n)$. Let's consider now the linear map $Y = A X$ where $Y \in \mathbb{R}^m$ with $m < n$. I am seeking $p(y_1,\ldots,...
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1answer
46 views

Multivariate conditional entropy

I would like to take data columns and compute the multivariate conditional entropy. For instance, suppose I have columns $A, B, C D, E$ and I want to compute the conditional entropy $H(E | A,B,C,D)$. ...
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1answer
47 views

Multivariate distribution: calculate P(Y > b/2)

The joint probability function looks like this: The first step for calculating $P(Y > 2/b)$ is calculating $f_Y(y)$. Which I did like this: The problem here is that my x is still in my indicator,...
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1answer
23 views

Understanding this expression of the multivariate t-distribution

I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows $$ G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\ Z = N (...
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Multivariate $t$ ('mvt') - adjustment in emmeans in R

I am doing post-hoc comparisons of contrasts based on linear mixed models I built in R. I am using the emmeans package for the comparisons. One of the default ...
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53 views

multivariate normal distribution with mean vector 0 and covariance matrix Σ

I am newby in statistics and I have huge data with "p" variables and "n" samples. My data is a two dimensional matrix with "n" columns (each column is a sample) and "p" rows (each row is a variable). ...
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1answer
31 views

Unbiased estimator for Theta of a Normal Distribution

If $X_1,\ldots,X_n\sim \operatorname{iid} \operatorname N(\theta, \sigma^2)$, then verify that $\bar{X}_n$ is unbiased estimator for $\theta$ and that Cramer Rao bound is met? I am facing difficulty ...
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1answer
40 views

Finding joint probability distributions from marginal distributions

Question: I was solving test papers where I found this one. My doubt: I know to work with conditional probabilities and Jaccobian Transformation and part A and B can be done applying the above..But ...
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1answer
37 views

Mean and variance of probability density with multidimensional indicator function

I encountered the following question while studying machine learning: We are asked to calculate mean and covariance of a given probability density function $$p(x) = \frac{1}{16} \cdot 1_{0 \leq x_1 ...
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Product of two multivariate Gaussian distributions of different dimensions

I am computing the posterior in a multi-output Bayesian regressor. I assume the prior to be a matrix Gaussian distribution. I can write the prior and the likelihood as multivariate normal ...
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20 views

How to compare two multivariate datasets

I have two datasets consisting of 20 features, one set contains 50k records and the other 70k. I want to check if they are from the same population. Datasets contain discrete features and the ...
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2answers
52 views

How to compare a new measurement to an existing multivariate distribution?

I have a dataset that describes the position and rotation of an object at different points in time using four dimensions. I want to use this sample of observations to get a sense of what positions and ...
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86 views

Deriving predictive distribution

In Bayesian Regression, I am confused how to to get $f*$ and $\sigma*$, given $$y^∗ \mid \vec{y}\sim\mathcal{N}(f^∗ , σ^∗ )$$ $$ p(y^* \mid \vec{y}) = \int{p(y^* \mid \vec{w}) p(\vec{w} \mid \vec{y})...
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Equality of two multivariate normal CDF's

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
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1answer
67 views

multivariate Student's t distribution: intuition for non-independence?

Consider a multivariate Student's t distribution, with parameters $\nu$ (d.f.), $\mu$ (location) and $\Sigma$ (shape). Does anyone have a good intuition for the individual components not being ...
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1answer
58 views

Variance of sum of random vectors - a proof

For nonrandom matrices $A(rXk)$,$ B(rXm)$, and $c(rX1)$, how does one show that $$\newcommand{\Var}{{\rm Var}}\newcommand{\Cov}{{\rm Cov}}\newcommand{\*}{{\times}} \Var(AX+BY+c)=A\Var(X)A′+ B \Var(...
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20 views

Multivariate Test for Mean Equivalence

I am looking for an equivalence test for multivariate means (arbitrary or normally distributed). Any suggestions or hints in the right direction are appreciated.
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23 views

Expectation of expressions involving sample covariance matrix and inverse of covariance matrix

Let $y_{ij}$, $i=1,2,\cdots,n_j$ be a random sample from $N_p(\mu_j,\Sigma_j)$, $j=1,2$. Let $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$ $$S_j=\frac{1}{n_j-1}\sum_{...
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Check whether a random sample comes from an elliptical distribution?

How can I check whether it is a reasonable assumption to say that a multivariate sample $x_1,...,x_n$ comes from an elliptical distribution, such as a normal distribution or a t-distribution? In the ...
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26 views

How to use Copulas to Combine Multivariate Conditional Probability with Univariate Conditional Probability?

This is sure to be an odd one, but here goes. I'm trying to estimate P(X|Y, Z) by the distributions of P(X|Y) and P(X|Z). I've thus far been trying to using copulas to achieve that aim, but I'm not ...
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1answer
58 views

A Multivariate Distribution for Linear Combinations of Independent Exponential Random Variables

Consider a random vector $\mathbf{X} \in \mathbb{R}^r$ whose components $X_j$ are independent exponential variables with different scale parameters $\beta_j$, $j=1,\dots,r$. Suppose I have a general $...
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44 views

Mutual information between multivariate random variables? [duplicate]

I have read that mutual information only works with two random variables, and that for 3 or more RVs there seems to be a variety of different measures (synergy, partial information decomposition, and ...
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1answer
65 views

Conditional distribution of multivariate Rayleigh distribution

The correlated Rayleigh envelopes using a set of zero-mean complex Gaussian RVs (Random Variables) is given by $$G_{k}=\sigma_{k}(\sqrt{1-\lambda_k^2}X_k+\lambda_kX_0)+i\sigma_{k}(\sqrt{1-\lambda_k^2}...
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102 views

Is multivariate Cauchy stable?

I am trying to prove (if possible), for given $A_{n\times n}$ and $B_{n\times n}$, there exists a $C_{n\times n}$ satisfying $$A\pmb{X}_1 + B\pmb{X}_2 \stackrel{D}{=} C\pmb{X},$$ where $X_1, ~X_2$, ...
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3answers
87 views

Hessian of Log of Matrix-t distribution

I am trying to calculate the hessian of the log of the matrix-t distribution. I know that the log of the matrix-t distribution can be written: $$\log T_{N\times P}(X| \nu, M, \Sigma, \Omega) \propto -\...
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Bayesian updating of MVN with joint signal

Suppose t and l are independent variables that are normally distributed with known means and variances. Suppose I get a signal s = f(t, l), where I know f(.) which is deterministic. How do I define ...
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1answer
72 views

If $\mathbf{x} \sim N(\mathbf{0,I})$ and $\mathbf{y} = \mathbf{Ax}$, what does $\mathbf{A}^T \mathbf{A}$ represent?

If $\mathbf{x} \sim N(\mathbf{0,I})$ then $\mathbf{AA}^T$ is the covariance matrix of $\mathbf{y} = \mathbf{Ax}$, but what does $\mathbf{A}^T \mathbf{A}$ represent? In some places I have seen ...
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15 views

Multivariate goodness-of-fit diagnostics

In a multivariate setting, we can assess the goodness-of-fit of a $p$-dimensional multivariate distribution to a set of $p$-dimensional multivariate data. Using, for example, the squared Mahalanobis ...
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Joint distribution of sample means from multivariate normal with unknown covariance matrix

The distribution of a (standardised) sample mean from a univariate normal distribution with unknown variance is given by the Student's t-distribution (Student's t-distribution). What about a joint ...
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1answer
145 views

MGF of the multivariate hypergeometric distribution

Does the multivariate hypergeometric distribution, for sampling without replacement from multiple objects, have a known form for the moment generating function?
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Multivariate stable distribution

I know that if $\pmb{X}_1$ and $\pmb{X}_2$ are independent copies of a $n \times 1$ random vector $\pmb{X}$, then $\pmb{X}$ is said to be sum stable in $\mathbb{R}^n$ if $a\pmb{X}_1 + b\pmb{X}_2 \...
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How does one choose a random isotropic direction and then have the vector have norm 1? [duplicate]

I want to choose a random vector in high dimensions such that it all directions have the same uniform chance (i.e. isotropic in all directions). My current idea is the following algorithm: sample v ...
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21 views

Mutual information for independent subsystems with non-Gaussian distributions

I seem to have found contradiction when computing mutual information for a multivariate Cauchy distribution. Below, I list a few things that I think are true. But I expect that at least one of them ...
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45 views

A question about the Multivariate Normal Distribution

From marginal of X1, X2 the first two elements of the mean vector is 0, 0. But I cannot find a well explianed process to find it..and I cannot find the dispersion matrix.. If someone helps me finding ...
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1answer
185 views

Joint distribution of independent t-distributed random variables

The multivariate t distribution seems to be defined as a "ratio" of a vector of normal random variables and a single gamma (or chi-squared) random variable (independent from the vector of normal). ...
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131 views

Can we apply a constraint on the distribution of the layer output?

As far I understood, the hidden layer outputs can be anything based on the learning algorithm or optimization rules. I was wondering if it possible to some constraints on the layer output. For ...
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23 views

A problem on Multivariate analysis

Firstly, if the random vector X follows trivariate normal, does it mean X1, X2, X3 follows normal ? I also donot know the formula of Multiple correlation coefficient P(suffix 2.13) I donot ...
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73 views

Transforming a multivariate binary time series to be stationary

I have a multivariate (multi-response) dataset with, for example, 10 different binary responses. I'm interested in an AR(p) model, determining how the responses at previous time steps relate to the ...
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27 views

Categorical probability distribution that captures “some” permutation invariance / mirror symmetry

I'm fitting something similar to a naive Bayes model to a data set where each data point has six features, $A_1$, $B_1$, $C_1$, $A_2$, $B_2$ and $C_2$. $A_1$ and $A_2$ can both take values in {$a_{1}$,...
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1answer
177 views

Correlation matrix for multivariate Cauchy distribution

I have found an equation for the entropy of a $p$-variate Cauchy distribution here [page 70]: $H(X,R) = \frac{1}{2}\log(\det(R))+f(p)\,,$ where $X=(X_1,X_2,\dots,X_p)$ is vector of random variables ...
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17 views

Deriving the semiaxes for the contour of a multivariate normal distribution

Given a mvd, we have that $(X-\mu)\Sigma^{-1}(X-\mu)^\top = c^2$ where $c^2 = \chi^2(\alpha)$ The semiaxes are supposedly $\pm c\sqrt{\lambda_i}p_i$ where $\lambda_i$ and $p_i$ are the eigenvalues and ...
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1answer
88 views

Why do I need multivariate normality tests?

I am new to time series analysis and would like to test a multivariate time series (12 components) for normality. I found several straightforward normality tests and some multivariate normality tests. ...
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0answers
10 views

Testing inequality restrictions on posterior mean

I want to compare to models in a multivariate regression using the posterior odds ratio in bayesian econometrics. I derived the posterior $p(\beta, h|y)$, that has mean $\beta$ and precision $h$, data ...
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1answer
89 views

finding Pr(X+Y > 500) given the following joint probability density function

I am reviewing for a probability and statistics class. I am stuck on a problem despite repeated attempts. (THIS IS NOT HOMEWORK!) The questions is: Consider an electronic system with two components. ...
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1answer
54 views

Show that $(\mathbf{x}, \boldsymbol\Theta\mathbf{x}+\boldsymbol\eta)$ is jointly normal

This is from Theodoridis' Machine Learning, exercise 3.16. Suppose $\mathbf{x}$ is a vector of jointly normal random variables with covariance matrix $\boldsymbol\Sigma_x$. Let $$\mathbf{y} = \...
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46 views

Why is conditional independence more important than marginal independence?

Graphical models are based on the idea of representing certain types of conditional independences in a (joint) distribution via a graph, and are an active research area. As argued (correctly I ...
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3answers
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What's the difference between Multivariate Gaussian and Mixture of Gaussians?

What's the difference between Multivariate Gaussian and Mixture of Gaussians? If I have a Multivariate Gaussian and making all the data into ONE vector, is that a Mixture of Gaussians in 1 dimension?...