# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 (...
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 $$\mu=[\mu_x, \mu_y]^T$$ $$\Sigma=\begin{bmatrix}... 0answers 31 views ### What is the entropy of multivariate data multiplied by a vector? It is a general rule that for multivariate data \boldsymbol{X} and a matrix \boldsymbol{A}, their entropy is$$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}| (...
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 $S_1$, ...