Questions tagged [multivariate-distribution]

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

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n-th quantile for bivariate variable

I generate a 2000 bivariate random samples which are negative correlated. I used np.quantile to generate 10 quantile from this random samples. The related point is marked in the following figure. I am ...
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t-Copula MLE on nu (DoF) only - log-likelihood function possibly convex?

I am working with t-Copula's to generate random synthetic data eventually. The paper I use as the foundation is Benali et al., 2021. To determine the best fitting t-Copula, they propose determining ...
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Overlap coefficient for two multidimensional normal distributions

For two PDFs $f_1(x)$ and $f_2(x)$ the overlap coefficient (OVL) measures the similarity between two distributions through the overlapping area of their distribution functions and is given by the ...
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Analogous result to Isserlis' theorem for mixed absolute product-moments of multivariate normal distribution

Suppose that $(X_1, \cdots, X_n)$ have a joint normal distribution. If $n = 2m + 1$, then $\mathbb{E} \left[ \prod_{j=1}^n X_j \right] = 0$. This can be argued from the symmetry of the multivariate ...
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Distribution of $X'\Sigma^{-1}X$ for $X$ following a multivariate $t$ distribution

According to Golam Kibria & Joarder (2006, p.7) available here and Kotz & Nadarajah (2004, p. 19) visible in google, the distribution of $X'\Sigma^{-1}X /p$, for a known correlation matrix $\...
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Which is the noncentral multivariate t distribuiton

In various documentations, I found out two definitions of the multivariate noncentral student $t$ distribution. The most commonly density (PDF) found is (e.g., wikipedia): $$\mathcal{T}(x;\mu, \sigma^...
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conditional expectation of random vectors

For a random vector $X=(X_1, ..., X_n)^\intercal$ the expectation value can be written as $\mathbb{E}[X] = (\mathbb{E}[X_1], ..., \mathbb{E}[X_n])^\intercal$ according to equation 2 in https://en....
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Expectation of a multivariate random variable

Given a multivariate random variable $\mathbf{X}=(X_1, ..., X_n)^\intercal : \Omega \rightarrow \mathbb{R}^n$ I want to determine the expectation value of this RV. Now wikipedia says the expectation ...
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Least likely sample in Multivariate Hypergeometric distributions

Let $U=(U_1,U_2,\dots,U_c)$ being an urn with $U_i$ balls of color $i \in [1,c]$ and $\Omega(U)$ be the set of possible draws from urn $U$. For $D \in \Omega(U)$ the probability of drawing $D$ is ...
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Finding density function of random vector (density transformation)

Let $X_{i} \sim \operatorname{Ga}\left(\alpha_{i}, \lambda\right)$ independent with $\alpha_{i}>0$ fūr $i=1,2$. Furthermore it is known, that $V=X_{1}+X_{2} \sim \operatorname{Ga}\left(\alpha_{1}+\...
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Expected value of inverse random variable with norm of multivariate normal

I need to calculate or at least estimate the expected value of this term $$ \mathbb{E} \left[\frac{1}{(L-K \cdot\left \| \mathcal{N}(0,\Sigma) \right \|_2)^2}\right] $$ where $L,K\in \mathbb{R}>0$ ...
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Understanding conditional distribution and sampling for dependent variables

Ok, so I am trying to see if I understand the following correctly. Suppose I have a bivariate random variable $(X,Y)$.. Just as an example, suppose that $X,Y$ is distributed as: $$\begin{align} Y&\...
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Multivariate Chebyshev's inequality with Mahalanobis distance

In Chebyshev's inequality, we can generalize the 68-95-99.7 rule from normal distributions to bound how much density is within a certain number of standard deviations from the mean. $$ P\big( \big\...
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From marginal distribution to joint distribution with independence

Consider a random vector $(X,Y,Z)$, Let $f_X, f_Y, f_Z$ be the probability distributions of each component. Question: Does there always exist a distribution $f$ for the whole vector $(X,Y,Z)$ such ...
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the mathematical representation for a generic multivariate distribution

I would like to use a formula represent a system as such, it has n variables, each variable is independent with each other; each variable should follow a discrete distribution respectively. In other ...
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Intuition behind redundant information

Suppose we have a set of variables, we call them "sources" ${\bf S} = \{S_1,\ldots,S_n\}$, and a "target" variable, denoted with $Y$. A measure of overall dependency between ${\bf ...
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Can every multivariate distribution be expressed as a function of univariate distributions of the same random variables?

Can every multivariate distribution $p(X)$ of a multivariate random variable $X = [X_1, X_2, \dots, X_d]^{T} \in \mathbb{R}^d$, be defined as some function of univariate distributions on $X_i$? I ...
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Absolute of expected value of multivariate correlated Bernoulli

I am running some experiment where I draw samples from a multivariate Bernoulli distribution (in this case taking values -1 or +1) with a single correlation coefficient (i.e., same correlation for all ...
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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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1 answer
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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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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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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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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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5 votes
2 answers
136 views

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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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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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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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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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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3 votes
2 answers
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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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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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4 votes
1 answer
796 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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1 vote
1 answer
357 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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6 votes
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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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4 votes
1 answer
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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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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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4 votes
2 answers
154 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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