Questions tagged [multivariate-distribution]

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

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Distribution of a product of a matrix with a random matrix

Suppose we have the matrices $Z\in \mathbb{R}^{n\times n}$ and $X\in \mathbb{R}^{n\times d}$, such that each row $x_i\in\mathbb{R}^d$ is drawn i.i.d from a $N(0,\Sigma_{d\times d})$ distribution. ...
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Your class has 100 students and they have 5 elective courses to choose from. In each course, the proportion of students is equal in population

I am actually unable to understand the question and would appreciate it if someone can help with that. For the above problem statement, there are three questions that I need to answer Make a ...
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106 views

Variance of expectation vs. Expectation of variance

Can we compare $Var_X[E_Y(f(X,Y))]$ and $E_Y[Var_X((f(X,Y))]$ where $f()$ is any function of $X$ and $Y$ iid? I suspect $Var_X[E_Y(f(X,Y))]$ is the smaller one. Though I couldn't find a single counter ...
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How is this equation deduced?

These two equations are from the book Gaussian Process for Machine Learning. First we already have equation (2.8). $p(\mathbf{w}|X, \mathbf{y}) ∼ N (\frac1{\sigma_n^2}A^{−1}X\mathbf{y}, A^{−1})$ (2.8) ...
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Is sampling from a transformed pdf the same as transforming samples in the same way from the original pdf?

Suppose I have a pdf $f(\vec{v})$ over $\vec{v}$ and define $\vec{u}=T(\vec{v})$, i.e. a transformation of $\vec{v}$. Would sampling $f(\vec{u})$ be the same as taking samples $\vec{v}_i$ from $f(\vec{...
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Are the mean of a joint distribution and its marginals equal?

I know that for a multivariate normal distribution the mean of the parameters is the same as the mean of the marginal distributions for the respective parameters. But is this always the case?
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28 views

How to apply the multivariate Poisson distribution?

From my recent question here: Calculating the probability of a car accident I wish to learn how to apply a multivariate poisson distribution in the example that car-accidents vary for each year, ...
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If normally distributed random vector $X$ has a PD covariance matrix, then any conditional distribution induced by $X$ has a PD covariance matrix?

Suppose I have a random vector $X$ whose distribution is joint normal. I know that the covariance matrix of $X$ is positive definite. I wonder if I partition $X$ in any way: e.g., $X=(X_1, X_2)$, then ...
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Mahalanobis distance?

I have a question regarding the use of the Mahalanobis distance as a measure of the goodness-of-fit of a bivariate Guassian pair. Let's assume that we have some theoretical relationship $$\bigg(\begin{...
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Restricting Kernel Density Estimate to n-sided polygon

Given some arbitrarily distributed data, in this case data generated by two normal distributions, $\mathcal{N_1}(\mu_1, \Sigma_1)$ and $\mathcal{N_2}(\mu_2, \Sigma_2)$ where $\mu_1 = \begin{bmatrix}0 &...
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quantile surface of a mulitvariate distribution made of multiplication of marginal distributions assuming independence

How to perform quantile regression in a more elegant fashion? As discussed above, quantSheets() can only deal with one explanatory variable for computing quantile ...
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Distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions

I am trying to simulate the distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions under the same covariance structure. Drawdowns are ...
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24 views

Sum of two different freedom degree multivariate t dsitributions

there are two multivariate t distributions whose freedom degree $\nu$ are different to each other. $$ \mathbf{x} \sim \mathcal{T}(\nu_x, \mathbf{0}, \Sigma_x) $$ $$ \mathbf{y} \sim \mathcal{T}(\nu_y, \...
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Testing underlying, latent normality of binary data

I have two multivariate binary datasets of 4 variables (or components) - P, Q, R and S each. Each row of the data is like a test on the components to say if they work (0) or fail (1) when tested. The ...
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How to show that $\left|\left|E\left[\frac{x-X}{||x-X||}\right]\right|\right|\xrightarrow{||x||\to\infty}1$ for a multivariate distribution?

I am trying to prove that $\left|\left|E\left[\frac{x-X}{||x-X||}\right]\right|\right|\xrightarrow{||x||\to\infty}1$ for any multivariate distribution. My Progress: I have proved the result for ...
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139 views

Mean of Generalization of the Dirichlet Distribution

I know that if $X_{1},X_{2},...X_{n}$ are independent $\mathrm{Gamma}(\alpha_{i},\theta)$ - distributed variables (notice they all have the same scale parameter $\theta$) and $Y_{i}=\frac{X_{i}}{\sum_{...
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60 views

Inner Product for Geometric Interpretation of Multivariate Random Vectors

I was looking into the geometric interpretation of random variables as random vectors in a vector space. The textbook I'm referring to defined $\operatorname {Cov}(X,Y)$ as the inner product for any ...
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35 views

PDF of a degenerate multivariable gaussian distribution

The pdf of a multivariate gaussian distribution with mean $\mu\in\mathbb{R}^n$ and variance $\Sigma\in\mathbb{S}^n_{++}$ ($\Sigma$ is positive definite) is given as $$ p(x) = \frac{1}{\sqrt{|\Sigma|(2\...
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Conditional distribution of cross covariance

Assume M follows an Inverse Wishart distribution (known parameters), what is the conditional distribution of X given Y and Z in the below (where Y and Z are square sub-matrices)? $$ M = \...
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What is the sample size required to estimate a multivariate joint histogram?

The required sample size for estimating a multivariate joint histogram is something that I expect to depend on multiple factors such as the distributional properties of the data-generating process (e....
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29 views

Generate a multivariate normal vector

I'm working in R and I was wondering, let's say I want to generate a random vector $X \in \mathbb{R^p}$, with $X \sim N(0,I)$ where $I$ is the identity matriz in $\mathbb{R}^{p \times p}$. The ...
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26 views

Combining marginal bivariate probability distributions into a single multivariate one

Consider a set of 3 correlated random variables $X$, $Y$ and $Z$. I have calculated bivariate marginal distributions over any pairs of these random variables. That is, I know probability density ...
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How to determine if a multi-dimensional vector comes from a multi-variate distribution or not?

I have a complex multivariate distribution given by a multivariate random number generator (black box). In other words, whenever I call the "generator" I get a random vector containing ...
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36 views

Simplifying assumption for regular vine distributions

I am trying to understand vine copulas and specifically, conditional bivariate copulas. Let $c_{X_1,X_3|X_2}(\cdot,\cdot | \cdot)=:c$ be the probability density function of a bivariate conditional ...
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Can a random variable be uncorrelated with its product with a correlated random variable?

I have a random variable $X.$ I want to find a random variable $Y$ such that $Y$ is correlated with $X,$ but $Y$ is not correlated with the product of $X$ and $Y.$ Is it always possible?
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Conditional Expectation of a normal distribution [duplicate]

say we have a multivariate normal distribution with ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$ The conditional expection is $\overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\...
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27 views

Test goodness of fit of a multivariate distribution via testing GOF of linear combinations of components

I am modeling a continuous bivariate distribution of a random vector $(X_1,X_2)$ using a copula. I would like to assess how well I am doing. Given a data sample, I could probably do a bivariate ...
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422 views

KL divergence for joint probability distributions?

I have a pair of joint probability distributions. I want to measure their similarity/dissimilarity. If they were single-dimensional probability distributions, then I could measure the Kullback–Leibler ...
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167 views

What is a generalisation of t-test to the case of multivariate distribution?

When we have a sample of numbers, one of the most basic tests is the t-test, in which we check the null hypothesis that the population mean is equal to zero. I am interested in a generalisation of ...
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Mutual information relationship to copula entropy is borked?

I have posted a related Question based on a paper, Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-54. In the paper, they ...
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Copula entropy: calculation is borked?

I came across a pretty cool paper whose idea makes a lot of sense to me. Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-...
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What is the best way to sample points from an arbitrary 2D distribution?

I want to sample points $(x,y)$ randomly according to the Himmelblau function $$f(x,y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2\qquad -5\le x,y\le 5$$ which I treat as a multivariate probability density ...
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Covariance matrix in multivariate is Wishart random variable

I'm working on sample variance distribution, on page 111 of Methods of Multivariate Analysis, it says "The joint distribution of these $p(p + 1)/2$ distinct variables in $W =(n−1)S = \sum_i(y_i − ...
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Is it reasonable to look at the output of simulating from a multivariate distribution as univariate distribution? If yes, what is this called?

Suppose I have $X_{n} \sim MVN(\underline{\mu},\Sigma)$ where $n$ is large (several thousands). However, the $\mu_i's$ and the elements of $\Sigma$ are such that almost every simulation from $X_n$ ...
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Measuring entropy/concentration in multivariate datasets

I am interested in measuring how concentrated or widely dispersed a certain binary property is among a population, which is defined by a number of categorical variables. For instance, let's say I have ...
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Multivariate Bernoulli Logistic Regression on OpenBUGS

I am new to WinBUGS/OpenBUGS, and am trying to solve this Multivariate Bernoulli Logistic Regression $$P(θ1,θ2|X,Z) ∝ P(X|θ1) P(θ1) P(Z|X,θ2) P(θ2)$$ Is this valid on OpenBUGS? This is my OpenBUGS ...
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Does the t-copula capture serial nonlinear dependence?

Say $\mathbf{Y}=[y_1,\cdots,y_n]'$ with elements that are uncorrelated, yet serially dependent. As a result, in the literature, the means of allowing for nonlinear serial dependence for processes ...
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PDF for Bayesian Decision

I have the discrete-time sequence $X(n) = 1$ with probability 3/4 and $X(n) = 0$ with probability 1/4 for $n = 1,...,10$. The observed signal is $Y(n) = X(n) + N(n)$ where $N(n)$ is an AWGN with ...
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139 views

Interpretation of multivariate conditional gaussian function form?

I've been reading over this Multivariate Gaussian conditional proof, trying to make sense of how the mean and variance of a gaussian conditional was derived. I've come to accept that unless I allocate ...
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Kth Order statistic of a multivariate distribution

The Kth order statistic for a univariate is equal to its kth-smallest value. For instance, given $\{6,9,3,8\}$, the 2nd-smallest value would be the 2nd order statistic. How does this concept ...
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53 views

Multivariate Gamma parameter estimation

Consider $X$ a d-dimensional random variable with positive values, mean $\mu\in\mathbb{R}_+^d$, and covariance matrix $\Sigma\in\mathbb{R}^{d\times d}$. If I have $n$ samples $\{ x_1, ..., x_n \}$ ...
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Derive expectation of the log determinant of a precision matrix from a Wishart distribution

I'm reading through section 21.6 of Murphy's Machine Learning: A probabilistic perspective where they derive the variational bayes algorithm for fitting a mixture of gaussians. One of the steps ...
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plotting bivariate normal distribution samples

I have generated two samples $\underline{X_i}$ and $\underline{Z_i}$ $\underline{X_1}$ $\underline{X_2}\dots \underline{X_{5000}}$ , while $\underline{X_i} \sim N_2[(1,2)^T,\begin{pmatrix}2&1.5\\ ...
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Why is there covariance instead of simply variance in multidimensional normal distributions?

Maybe it is just because I'm only experimenting with 2 dimensional normal distributions, but multi-dimensional normal distributions for me seem like just multiple one dimensional normal distributions. ...
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Probability distribution associated with nuclear norm?

The $\ell_1$ norm of model parameters is often added to loss functions because it induces sparsity in the solution of the overall cost function: $$ c(\theta) = \log L(x|\theta) + \lambda ||\theta||_1$$...
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Condition Random Variable on Range of Another Random Variable [duplicate]

Assume that $v_s \sim N(\mu_s,\sigma_s^2)$ and $v_b \sim N(\mu_b,\sigma_b^2)$, denote their correlation by $\rho$, and assume they are jointly normally distributed. How would I assess $E[v_b|v_s\leq c]...
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Bivariate random effect problems in selection models (Mixture Cure model)

I am currently working on a mixed effects selection model. The selection model is a logistic model with a Gaussian random effect. The principal model is a survival model with a Gaussian random effect (...
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168 views

Find expectation of conditional normal distribution

I am struggling with some finding expectation value question . the question is to find $E[Y|X]$ from the result $P(Y|X)$ with given mean and covariance $$\mu=[\mu_x, \mu_y]^T$$ $$\Sigma=\begin{bmatrix}...
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What is the entropy of multivariate data multiplied by a vector?

It is a general rule that for multivariate data $\boldsymbol{X}$ and a matrix $\boldsymbol{A}$, their entropy is $$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$ (...
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What is the distribution of the CDF of a sample drawn from a multivariate normal?

Introduction: Lets say we have a random variable $X$ that follows a normal distribution, $X \sim N(\mu, \sigma^2)$ , with a CDF function $F_X(x) = P(X \leq x)$. Then we draw some random samples $S_1$, ...