# Questions tagged [multivariate-distribution]

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

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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 ...
1 vote
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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 ...
23 views

### 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 ...
27 views

### 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. ...
385 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 ...
28 views

### 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 ...
193 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 ...
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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 $P(XY \leq t)$. ...
1 vote
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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 ...
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 ...
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1 vote
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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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1 vote
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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....
2k views

### 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 ...
53 views

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}+\... 3 votes 0 answers 145 views ### 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$L,K\in \mathbb{R}>0$... 0 votes 1 answer 63 views ### 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$X,Yis distributed as: \begin{align} Y&\... 4 votes 1 answer 172 views ### Multivariate Chebyshev's inequality with Mahalanobis distance In Chebyshev's inequality, we can generalize the 68-95-99.7 rule from normal distributions to bound how much density is within a certain number of standard deviations from the mean. P\big( \big\... 3 votes 1 answer 107 views ### From marginal distribution to joint distribution with independence Consider a random vector(X,Y,Z)$, 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 ... 3 votes 0 answers 29 views ### 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 ...
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 ...
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. ...