Questions tagged [multivariate-distribution]
Probability distribution over vectors (as opposed to univariate distributions that are over numbers).
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10 views
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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0answers
6 views
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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0answers
15 views
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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0answers
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
$$
G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\
Z = N (...
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18 views
Multivariate $t$ ('mvt') - adjustment in emmeans in R
I am doing post-hoc comparisons of contrasts based on linear mixed models I built in R. I am using the emmeans package for the comparisons. One of the default ...
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53 views
multivariate normal distribution with mean vector 0 and covariance matrix Σ
I am newby in statistics and I have huge data with "p" variables and "n" samples. My data is a two dimensional matrix with "n" columns (each column is a sample) and "p" rows (each row is a variable). ...
0
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1answer
31 views
Unbiased estimator for Theta of a Normal Distribution
If $X_1,\ldots,X_n\sim \operatorname{iid} \operatorname N(\theta, \sigma^2)$, then verify that $\bar{X}_n$ is unbiased estimator for $\theta$ and that Cramer Rao bound is met?
I am facing difficulty ...
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1answer
40 views
Finding joint probability distributions from marginal distributions
Question:
I was solving test papers where I found this one.
My doubt:
I know to work with conditional probabilities and Jaccobian Transformation and part A and B can be done applying the above..But ...
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1answer
37 views
Mean and variance of probability density with multidimensional indicator function
I encountered the following question while studying machine learning:
We are asked to calculate mean and covariance of a given probability density function
$$p(x) = \frac{1}{16} \cdot 1_{0 \leq x_1 ...
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0answers
29 views
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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0answers
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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0answers
86 views
Deriving predictive distribution
In Bayesian Regression, I am confused how to to get $f*$ and $\sigma*$, given
$$y^∗ \mid \vec{y}\sim\mathcal{N}(f^∗ , σ^∗ )$$
$$
p(y^* \mid \vec{y}) = \int{p(y^* \mid \vec{w}) p(\vec{w} \mid \vec{y})...
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0answers
46 views
Equality of two multivariate normal CDF's
Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
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1answer
67 views
multivariate Student's t distribution: intuition for non-independence?
Consider a multivariate Student's t distribution, with parameters $\nu$ (d.f.), $\mu$ (location) and $\Sigma$ (shape).
Does anyone have a good intuition for the individual components not being ...
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1answer
58 views
Variance of sum of random vectors - a proof
For nonrandom matrices $A(rXk)$,$ B(rXm)$, and $c(rX1)$, how does one show that
$$\newcommand{\Var}{{\rm Var}}\newcommand{\Cov}{{\rm Cov}}\newcommand{\*}{{\times}}
\Var(AX+BY+c)=A\Var(X)A′+ B \Var(...
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0answers
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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0answers
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 $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$
$$S_j=\frac{1}{n_j-1}\sum_{...
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0answers
38 views
Check whether a random sample comes from an elliptical distribution?
How can I check whether it is a reasonable assumption to say that a multivariate sample $x_1,...,x_n$ comes from an elliptical distribution, such as a normal distribution or a t-distribution?
In the ...
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0answers
26 views
How to use Copulas to Combine Multivariate Conditional Probability with Univariate Conditional Probability?
This is sure to be an odd one, but here goes.
I'm trying to estimate P(X|Y, Z) by the distributions of P(X|Y) and P(X|Z). I've thus far been trying to using copulas to achieve that aim, but I'm not ...
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1answer
58 views
A Multivariate Distribution for Linear Combinations of Independent Exponential Random Variables
Consider a random vector $\mathbf{X} \in \mathbb{R}^r$ whose components $X_j$ are independent exponential variables with different scale parameters $\beta_j$, $j=1,\dots,r$. Suppose I have a general $...
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0answers
44 views
Mutual information between multivariate random variables? [duplicate]
I have read that mutual information only works with two random variables,
and that for 3 or more RVs there seems to be a variety of different measures (synergy, partial information decomposition, and ...
1
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1answer
65 views
Conditional distribution of multivariate Rayleigh distribution
The correlated Rayleigh envelopes using a set of zero-mean complex Gaussian RVs (Random Variables) is given by
$$G_{k}=\sigma_{k}(\sqrt{1-\lambda_k^2}X_k+\lambda_kX_0)+i\sigma_{k}(\sqrt{1-\lambda_k^2}...
7
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0answers
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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0answers
17 views
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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0answers
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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0answers
62 views
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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52 views
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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0answers
54 views
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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0answers
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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0answers
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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0answers
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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0answers
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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0answers
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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0answers
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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0answers
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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0answers
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
1k views
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?...
