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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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How may I find the distribution of a transformation of multivariate random variables?

Forgive me if this question has already been asked on here, but I could only find posts if the multivariate random variables were multivariate Gaussian. Suppose we have two multivariate random ...
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Hotelling's $T^2$ chart for subgroups with unequal size

I have been reading about Hotelling's $T^2$ control charts and I'm unsure on how to deal with the case where the mean observations come from unequal-sized subgroups. Consider $m$ observations $\mathbf{...
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Optimizing parameters for a non-standard probability density function

We have a non-standard multivariate probability density function, P(x | q), where x is a vector, and q are the parameters of the density. I get events ...
Niteya Shah's user avatar
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PMF of the Independent Multivariate Bernoulli Distribution

I was reading this paper on the Multivariate Bernoulli Distribution, which provides the general form of the PMF in equation 3.1. The paper refers to this as the probability distribution function, but ...
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Sampling from multivariate probability distribution

What's the best way to sample from multivariate probability density functions that are proportional to $\exp(-\|x\|_2)$ or $\|x\|_2^p \exp(-\|x\|_2)$ for some positive integer $p$ with $x \in \mathbb{...
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3 votes
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Multivariate Normal Distribution. How do we apply this to dataset?

I am having a hard time understanding the concept of a multivariate normal distribution. From my understanding, it assumes each group is normally distributed, making one joint normal distribution with ...
Taewooo Kim's user avatar
1 vote
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Parametric copulas with marginals that are regressions

In Dependence Modeling with Copulas (Harry Joe) I'm struggling to interpret the meaning of a statement. In Chaper 5.1, it is stated: Parametric inference for copulas For dependence modeling with ...
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Multivariate sample covariance

I have a set of $X_1,...,X_n$ samples with covariance $\Sigma_1,...,\Sigma_n$. The multivariate sample mean is then $$ \left(\sum_{i=1}^n \Sigma_i^{-1} \right)^{-1} \left(\sum_{i=1}^n \Sigma_i^{-1} ...
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Product of Two t-distribution Formulas

Does the product of two t-distribution formulas with same degrees of freedom simplify? $T_v(x; \mu_1, \Sigma_1)T_v(x; \mu_2, \Sigma_2) =\ ?...$ In the normal case it simplifies to: $\mathcal{N}(x; \...
Snowy Baboon's user avatar
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Distribution of covariance parameter resulting from sum of known- and unknown-covariance noise processes

Let's say I have a set $X$ of $N$ random $p$-dimensional vectors generated by $\mathbf{x}_i = \boldsymbol{\mu} + \Psi_i^{1/2} \boldsymbol{\xi}_i + \Sigma^{1/2} \boldsymbol{\zeta}_i$, where $\...
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How do you count degrees of freedom in multivariate distributions?

When dealing with a multivariate distribution (e.g., a multivariate t-distribution), how does dimensionality play into degrees of freedom? Let's say you have N measurements of a d-dimensional vector, ...
hank's user avatar
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Generate multivariate distributions of lognormal and normal distribution in python

I need to generate random numbers from 3 correlated distributions. First two of them are lognormal and the final one is normal, i.e. for X, ...
Xu Shan's user avatar
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6 votes
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multicomp package and emmeans package produce different adjust pvalues for Dunnett procedure [closed]

For Dunnett adjustment, multicomp package and emmeans package in R give different results. Anyone knows why? Thanks. Please see ...
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Estimate multivariate distribution with several variables on real data (continuous and categoricals) and sample from it

I have a complex dataset, collected through a survey, with both continuous (such as Age, Body mass index, etc..) and categorical variables (i.e. Gender, Education, etc..). I want to estimate their ...
SchefSTAT's user avatar
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Difference between random vector and joint distribution

My definition of a random vector is a vector $(x_{1},...,x_{n})$ that maps from a sample space to ${R}^{n}$. An example of this (in my understanding) would be a random process such as drawing one card ...
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Multivariate and integral representations explanation

I would to understand more on the multivariate and integral representations. I am fascinated by the possibility of a multivariate distribution like this one: $$\mathbb{P}(X_1 \le Z + a, X_2 \le Z + a, ...
Francesco's user avatar
1 vote
1 answer
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Calculation of multivariate probability mass function

How to calculate the following multivariate probability mass function: $P(X_1-X = n, X_2-X = n, ..., X_{N-1}-X = n)$ Where $n$ and $N$ are positive integers, and $X_i$ and $X$ are iid random variables ...
Francesco's user avatar
3 votes
1 answer
95 views

Quantile function of multivariate distributions from empirical samples

Let's say I have a k-dimensional multivariate normal distribution $MVN(0,\Sigma)$. Denote random vector $X \sim MVN(0,\Sigma)$ as $X = (x_1, x_2, \dots, x_k)$. It is trivial to find $P(x_1 \leq c_1, ...
David Wang's user avatar
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Approximating a bivariate distribution with another distribution, which method to use?

Let $X \sim F(;\theta)$ and $Y \sim G(;\eta)$ be two independent continuous random variables. The greek symbols represent the parameters of those distributions. I can easily sample from these ...
Coolio's user avatar
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Find the conditional PDF of a multivariate normal distribution given a constraint [duplicate]

Problem to solve We have a vector of random variables $\textbf{X}=(X_1,X_2)$ issued from a bivariate normal distribution. In particular, $\mu = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$, $\Sigma = \begin{...
Cairoknox's user avatar
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Joint testing of proportions, analogous to the Hotelling $T^2$ test?

Hotelling's $T^2$ test examine if two multivariate Gaussian distributions with the same covariance matrix have the same mean vector. Assuming the two distributions to have the same covariance matrix ...
Dave's user avatar
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1 vote
1 answer
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Standard deviation of a function of random variables

I was a bit confused recently with why two jointly distributed variables will exhibit a smaller scatter as the correlation between the random variables increases (assuming only positive values for the ...
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Copula Invariance Principle

I don't get why equation 7 is true, can someone explain me why? This is part of the proof of the invariance principle in copula theory.
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Estimating the sample variance and expected value of a maximization function of multiple random variables

Suppose I have three i.i.d. random variables $X$, $Y$, and $Z$. A model is used to generate $i$ outcomes for each variable that are used to estimate the sample mean and sample variance for each of the ...
C Greene's user avatar
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Recreating Heffernan-Tawn Fig. 6 in R: samples from multivariate extreme value distribution

I am trying to recreate Figure 6 from Heffernan-Tawn (2004) A conditional approach for multivariate extreme values using the same datasets as used in the paper. The original time series data of air ...
ThreeOrangeOneRed's user avatar
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86 views

Probability distribution of the product of three dependent continuous random variables

Given the joint probability distribution for three dependent continuous random variables, I want to find a formula to compute the probability distribution for the product of these 3 random variables. ...
MadMax2048's user avatar
9 votes
1 answer
611 views

Kullback–Leibler divergence between two multivariate t distributions with different degrees of freedom?

I want to calculate the Kullback–Leibler divergence between two multivariate $t$ distributions with different degrees of freedom (say $\nu_1$ and $\nu_2$), but same location and scale matrix, for ...
Student's user avatar
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4 votes
1 answer
205 views

Generate nonnegative variates with mean 1 and specified variance-covariance

Problem In several applications in surveys, it would be helpful to be able to generate a set of $R$ $n$-dimensional variates with the following properties: Has mean vector $1$ Has a specified ...
bschneidr's user avatar
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Statistic for significance test comparing transition matrix to null

I have several pitch sequences (mini songs), that look something like this when plotted as a pitch profile (piano roll): I've coded each sequence in terms of its constituent interval sizes, and ...
z8080's user avatar
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1 vote
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Measure the "actual" number of dimentions in a multivariate distribution

Consider a 3D multivariate normal distribution $x\sim N(0,\Sigma)$ where $$\Sigma=\begin{bmatrix}1 &1 &0 \\ 1&1&0 \\ 0 &0& 1 \end{bmatrix}$$ Since $x_1$ and $x_2$ are fully ...
elemolotiv's user avatar
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2 votes
0 answers
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Test for multivariate stationarity

Given a bivariate time series $(X_{1t}, X_{2t})$ with $X_{it}\sim N(0,1)$, I want to know if there is a way of testing if the bivariate distribution does not change over time. This is, if $F_{12}(x_1,...
Jesús A. Piñera's user avatar
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0 answers
31 views

Distribution of iid variates with fixed sum

I have $n$ bins, and a pool containing $N$ balls, $N > n$. I do a consecutive $N$ times: I take one of the balls from the pool and put it into a bin randomly selected with uniform probability $1/n$....
Sakuragaoka's user avatar
1 vote
1 answer
164 views

Creating multivariate distribution using marginal: can I use a copula?

I am modeling a multivariate distribution, where I already have the distributions of the marginals. Let's call the marginals, $f_{i}(x)$ for every $i$ marginal distribution, where $i=1,2,3$. We ...
deblue's user avatar
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119 views

Contour lines for multivariate Laplace distribution

In the p-variate normal distribution ($\mathbf{N_p(\mu,\Sigma)}$), the solid ellipsoid of x values satisfying $(\mathbf{x-\mu)'\Sigma^{-1}(\mathbf{x}-\mu)}\leq \chi^2_p(\alpha)$ has probability $1-\...
Helder Alves Arruda's user avatar
2 votes
0 answers
122 views

When to specify multivariate versus univariate priors on parameters?

Suppose a linear regression model: $$y \sim Normal(\beta X, \sigma)$$ For our purposes, assume $y$ is a univariate outcome and $X$ is a design matrix containing an intercept and one additional ...
socialscientist's user avatar
2 votes
1 answer
127 views

Joint Uniform Distribution Probability Problem

Let $X \sim U(0,1) $ and $Y \sim U(0,x) $. Calculate $$ \Pr(X >0.5 | Y= 0.25)$$ Is this a trick question ? Since $\Pr(Y = 0.25) = 0$, right ?
actsci stud tries2learn math's user avatar
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0 answers
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Mistake in Multivariable Transformation

The answer is $\frac{19}{24}$ but I don't know where is my mistake. The question is this. Let $$f_{X,Y}(x,y) = \begin{cases} x+y, \quad 0<x<1, 0<y<1 \\ 0, \quad \quad \quad \text{elsewhere}...
actsci stud tries2learn math's user avatar
1 vote
1 answer
207 views

Sampling from multivariate Bernoulli

Suppose you have a vector p drawn from a multivariate Beta distribution (not a Dirichlet), such as the one described here ( How to construct a multivariate Beta distribution? ) with a Gaussian copula. ...
fm361's user avatar
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1 vote
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Why is it difficult to sample from a multivariate distribution? [closed]

The Monte Carlo Markov Chain method requires sampling from a multivariate distribution. This is because the Markov Chain process requires dependent draws. See 1:55 at https://www.youtube.com/watch?v=...
Snoopy's user avatar
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4 votes
0 answers
152 views

How to fit a copula when zeros abound?

I am modelling a joint distribution for two random variables: $F(x,y)$. I observe $n$ data points $(x^{}_{i},y^{}_{i})^{N}_{i=1}$. I would like to model $F$ as the product of its marginals and a ...
lasoon's user avatar
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2 votes
1 answer
597 views

Fitting a copula vs. directly fitting a multivariate distribution

I understand that the joint density of two random variables $f(x,y)$ can be decomposed as the product of its marginals and a copula: $f(x,y) = g(x)k(y) \times c(G(x),K(y))$. Alternatively this may be ...
lasoon's user avatar
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0 answers
131 views

Probability of getting k different colored balls from an urn with K different colors of balls, each color has the same number of balls

Let's say I have an urn with $n$ balls, with $K$ different colors of balls, where each color has the same number of balls: $\frac{n}{K}$. Given I reach into the urn and grab a ball (without ...
Quantum Guy 123's user avatar
6 votes
1 answer
187 views

What is the distribution of a random linear combination of gamma random variables?

Let $U\sim \mathcal U(0, 1)$ be a random variable uniformly distributed over the interval $[0, 1]$. Let $X_1, X_2\sim \Gamma(a, b)$ be two iid random variables with a Gamma distribution. Now it is ...
jfiedler's user avatar
0 votes
0 answers
53 views

generalization of univariate pdf under constraints

I'm looking for generalizations of a univariate probability distribution function. The pdf is $$ \varphi(x)=(2\sqrt{s}K_1(2\sqrt{s}))^{-1}e^{\frac{s}{\log x}}. $$ for $K_1$ a modified Bessel function, ...
geocalc33's user avatar
4 votes
2 answers
336 views

Multivariate Probability Distribution with Linear Conditional Expectation

I want to know what probability distribution has the linearity property of the conditional expectation. To be specific, suppose that we have three random variables named $v_1,\;v_2,\;v_3$. Then, if $[...
5 votes
2 answers
223 views

Random variate of a singular Wishart distribution with non-integral degrees of freedom

Let's say that I have a random variable $X$ that follows a singular Wishart distribution with $\nu$ degrees of freedom and a shape matrix $\Sigma$ that is $p\times p$. We can write "symbolically&...
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1 vote
0 answers
152 views

Multivariate CDF with a Closed From

Is there any multivariate distribution with a closed form CDF? As we know, if $X_1,\;\ldots,\;X_M$ are i.i.d. type-1 extreme value distributed, $\Pr[X_2-X_1<c_1,\;\ldots,\;X_M-X_1<c_M]$ has a ...
M.C. Park's user avatar
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1 vote
0 answers
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Joint normality of a vector derived from joint normal vectors

Suppose that we have two random vectors following joint normal distributions: $$X=[x_1,x_2]'\sim N(0,\Sigma_X)\quad \textrm{and}\quad Y=[y_1,y_2,y_3]'\sim N(0,\Sigma_Y).$$ In this setup, I am ...
M.C. Park's user avatar
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7 votes
2 answers
657 views

How can I generate random observations from a concrete copula?

Let us assume that we have two continuous random variables $X$, $Y$, with known distributions (not necessarily normal), connected/related via a concrete copula. What is a procedure to generate random ...
Vicent's user avatar
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1 vote
1 answer
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How to calculate conditional probability on student multivariate distribution

I have a multivariate student distribution fitted on some data on 4 dimensions (so I know the parameters). I am trying to calculate the $P(X_4\le x_4| X_1=x_1, X_2=x_2, X_3=x_3)$ but falling short. I ...
Mayeul sgc's user avatar

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