Questions tagged [multivariate-distribution]
Probability distribution over vectors (as opposed to univariate distributions that are over numbers).
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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, ...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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Does marginalizing the covariance of a Normal (with a Wishart prior, not inv-Wishart) lead to a t distribution?
It's known that integrating out $\Lambda \equiv \Sigma^{-1}$ below,
$$ y|\Lambda \sim \mathcal N(0, \Lambda^{-1}), $$
$$ \Lambda \sim \mathcal W(M^{-1}, \nu) $$
leads to a multivariate t distribution ...
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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 ...
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Statistical Inference by Casella Exercise 4.51 [duplicate]
I am self-studying statistical inference by Casella and Berger and having difficulty solving exercise 4.51:
let $X, Y, Z \sim U(0,1)$ and they are independent. Find $P(X/Y \leq t)$ and $P(XY \leq t)$. ...
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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 ...
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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 ...
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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,...
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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 $1/n$....
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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 ...
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Multivariate Hypergeometric distribution for non-exchangeable sequence
I have a problem that I think is similar to the Multivariate Hypergeometric distribution urn problem as described on Wikipedia.
I have a dataset where each row is a trial. All trials have the same ...
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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-\...
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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 ...
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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 ?
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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}...
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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.
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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=...
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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 ...
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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 ...
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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 ...
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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 ...
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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}}. $$
for $K_1$ a modified Bessel function, ...
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Multivariate Probability Distribution with Linear Conditional Expectation
I want to know what probability distribution has the linearity property of the conditional expectation.
To be specific, suppose that we have three random variables named $v_1,\;v_2,\;v_3$.
Then, if $[...
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Random variate of a singular Wishart distribution with non-integral degrees of freedom
Let's say that I have a random variable $X$ that follows a singular Wishart distribution with $\nu$ degrees of freedom and a shape matrix $\Sigma$ that is $p\times p$. We can write "symbolically&...
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Multivariate CDF with a Closed From
Is there any multivariate distribution with a closed form CDF?
As we know, if $X_1,\;\ldots,\;X_M$ are i.i.d. type-1 extreme value distributed,
$\Pr[X_2-X_1<c_1,\;\ldots,\;X_M-X_1<c_M]$ has a ...
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Joint normality of a vector derived from joint normal vectors
Suppose that we have two random vectors following joint normal distributions:
$$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).$$
In this setup, I am ...
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How can I generate random observations from a concrete copula?
Let us assume that we have two continuous random variables $X$, $Y$, with known distributions (not necessarily normal), connected/related via a concrete copula.
What is a procedure to generate random ...
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How to calculate conditional probability on student multivariate distribution
I have a multivariate student distribution fitted on some data on 4 dimensions (so I know the parameters).
I am trying to calculate the $P(X_4\le x_4| X_1=x_1, X_2=x_2, X_3=x_3)$ but falling short.
I ...
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n-th quantile for bivariate variable
I generate a 2000 bivariate random samples which are negative correlated. I used np.quantile to generate 10 quantile from this random samples. The related point is marked in the following figure. I am ...
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Overlap coefficient for two multidimensional normal distributions
For two PDFs $f_1(x)$ and $f_2(x)$ the overlap coefficient (OVL) measures the similarity between two distributions through the overlapping area of their distribution functions and is given by the ...
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Distribution of $X'\Sigma^{-1}X$ for $X$ following a multivariate $t$ distribution
According to Golam Kibria & Joarder (2006, p.7) available here and Kotz & Nadarajah (2004, p. 19) visible in google, the distribution of $X'\Sigma^{-1}X /p$, for a known correlation matrix $\...
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Which is the noncentral multivariate t distribuiton
In various documentations, I found out two definitions of the multivariate noncentral student $t$ distribution.
The most commonly density (PDF) found is (e.g., wikipedia):
$$\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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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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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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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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