Questions tagged [random-vector]

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Are moduli of components of Fourier transformed Gaussian random vector still independent?

Suppose $X=[X_1,\ldots, X_n]^T$ is a random (column) vector such that: $X_i \stackrel{i.i.d}{\sim} \mathcal{N}(0,\sigma^2), \ 1 \le i \le n$ $\mathcal{F} \in \mathbb{C}^{n \times n}$ is the discrete ...
Adam O. G.'s user avatar
2 votes
1 answer
126 views

Representation of two Gaussian vectors as sums of independent Gaussian vectors

I know that if we have 1-dimensional Gaussian r.v. X and Y we can find coefficients $a,b$ so that $$X=aY+bZ+E[X-Y]$$ where $Y , Z$ are independent and $Z$ is standard Gaussian. Can we do something ...
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Multivariate Log-Normal variables with given covariance

Given a symmetric positive definite matrix $\bf \Sigma \in \mathbb{R}^{n \times n}$, I want to find a matrix ${\bf \Gamma} \in \mathbb{R}^{n \times n}$ and a vector ${\bf m} \in \mathbb{R}^n$ such ...
iLikeBayes's user avatar
3 votes
1 answer
118 views

What does it mean for a sequence of random vectors to converge to a random vector?

I am reading about convergence of random variables from Wikipedia and I come across this. Note that the condition that $Y_n$ converges to a constant is important, if it were to converge to a random ...
figs_and_nuts's user avatar
1 vote
0 answers
38 views

Visualization of $L(X)$

Let $L^2_+$ the set of all $2$-dimensional nonnegative random vectors $X = (X_1, X_2)^⊤$ with finite and positive marginal expectations, and let $Ψ^{(2)}$ the class of all measurable functions from $\...
geocalc33's user avatar
5 votes
1 answer
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 ...
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327 views

covariance matrix for three correlated variables

Suppose I have a covariance matrix for three random variable $X1,X2,X3$ $$ \begin{bmatrix} 1&0.5& \rho \\0.5&1&0.5 \\\rho&0.5&1 \end{bmatrix} $$ I know I can solve for valid ...
Pablitorun's user avatar
1 vote
0 answers
246 views

Variance of a vector function

One way of defining the variance of a vector is as follows \begin{align*} \text{Var}(g) = \mathbb{E}[ \, \lVert g \rVert_2^2\, ] - \lVert\,\mathbb{E}[ g ] \, \rVert_2^2. \tag{1}\label{1} \end{align*} ...
Taw's user avatar
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408 views

Covariance matrix for multivariate normal random variable

Suppose we have a multivariate normal random variable X = [X1, X2, X3, X4]^⊤ And here X1 and X4 are independent (not correlated) Also X2 and X4 are independent But X1 and X2 are not independent. ...
Rahul Singh's user avatar
2 votes
0 answers
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Median zero in the multivariate case

I have a question about the definition of the median of a (continuous) random vector $X\equiv (X_1,..., X_r)$. As suggested here and by other discussions in this forum (e.g., here and here) there are ...
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Let $\textbf{x}$ be some constant fixed vector, and $\textbf{y}$ a uniformly distributed vector, what is the distribution of the inner product?

Suppose that $\textbf{x}$ is fixed vector in $\mathbb{R}^n$, and let $\textbf{y}$ be a vector in $\mathbb{R}^n$ that is sampled according to a uniform distribution. Conditional on $\textbf{x}$ being ...
Link L's user avatar
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0 answers
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Help factoring matrices out of cross-covariance

I am trying to prove that $\text{Cov}( \boldsymbol{BU}, \boldsymbol B' \boldsymbol W) = \boldsymbol B \text{Cov}(\boldsymbol U, \boldsymbol W) \boldsymbol B'^T$, where $\boldsymbol B$ is $j \times m$...
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Probability that any element of a random unit-length vector is large [closed]

Given a vector $X \in R^n = \{x_1, x_2, ..., x_n\}$ drawn uniformly such that: $x_i \in [0, 1]$ for all $i$; and $\sum x_i = 1$, how would you find the probability that any of the $x_i > y$, for ...
RedPanda's user avatar
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3 votes
1 answer
233 views

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 ...
vikram71198's user avatar
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1 answer
239 views

Proving that a random vector is not bivariate normal

Suppose X,Y are random variables and their joint pdf is given by: f(x,y)=2g(x)g(y) where x*y>0, and zero otherwise. g(x) and g(y) are pdfs of standard normal distribution. I was first able to prove ...
Michael's user avatar
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1 answer
58 views

$E[X^T (Y-Z)] = E[X^T] E[Y-Z]$ but what about $E[(X^T (Y-Z))^2]$?

Let $X, Y$, and $Z$ be random vectors with $X$ independent of $Y$ and $Z$. Due to the independence we have $$ E[X^T (Y-Z)] = E[X^T] E[Y-Z]. $$ But what what $E[(X^T (Y-Z))^2]$? Is it possible to ...
Bertus101's user avatar
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7 votes
1 answer
110 views

Evaluating (Uniform) Expectations over Non-simple Region

Background. Let $V = (X,Y)$ be a random vector in 2-dimensions uniformly distributed over two disjoint regions $R_X \cup R_Y$ defined as follows: $$ \begin{align} R_X &= ([0,1] \times [0,1]) \...
EzioBosso's user avatar
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82 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 $$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$ (...
develarist's user avatar
  • 3,559
0 votes
1 answer
72 views

Correlation between two vectors sharing some elements

Suppose $\mathbf{x,y}$ are column vectors of length $n$, where $x_i,y_i \sim \mathcal{D}, \forall i \in \{1,2,\ldots ,n\}$. $\mathbf{z}$ shares $m$ elements with $\mathbf{x}$ and $n-m$ elements with $\...
hyiltiz's user avatar
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2 answers
398 views

Are two random vectors independent if their corresponding components are all independent?

Let $\mathbf{X} = (X_1,\ldots,X_n)$ and $\mathbf{Y} = (Y_1,\ldots,Y_n)$ be random vectors, and let $f_{\mathbf{X}}(x_1,\ldots,x_n)$ and $f_{\mathbf{Y}}(y_1,\ldots,y_n)$ be their respective pdfs or ...
Leonidas's user avatar
  • 121
7 votes
2 answers
487 views

Variance and asymptotic normality of $\frac{1}{n-1}\sum_{i=1}^{n-1}(x_{i+1}-x_i)^2$, where $X \sim \mathcal{N}(0,1)$

Consider a length $n$ vector $\mathbf{x}$ containing $n$ i.i.d. observations $\{x_i\}_{i=1}^n$ of a standard normal random variable $X$. Let $\mathbf{z}$ be a length $n-1$ vector whose entries are $...
eyeExWhy's user avatar
  • 556
4 votes
1 answer
338 views

Assume $X,Y$ are two independent random variables. Let $Z=f(X,Y)$. If $Z$ is independent of $X$, $f(X,Y)$ is constant in $X$. Is this true?

Let $X\in \mathbb{R}^n$ and $Y\in \mathbb{R}^m$ be two independent random vectors. Then, say that we have a third real valued random variable $Z=f(X,Y)$, with $f$ being measurable. Say that we know ...
random_name's user avatar
1 vote
1 answer
263 views

Hypothesis Test on the Difference between two random vectors

Each of my vectors consists of beta estimates for two separate models of the same data and the same number of explanatory variables. The question is asking whether the difference between these two ...
Alex's user avatar
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0 answers
477 views

How would you decorrelate a collection of vectors so that two vectors are uncorrelated?

Suppose $X_1, \ldots, X_K$ are all $\mathbb{R}^d$-dimensional random variables each with correlation matrix $\text{Var}(X_k) = \Sigma_{k} \in \mathbb{R}^{d \times d}$. Suppose we observe samples $X_{...
cgmil's user avatar
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1 answer
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If $cov(x_i,T_i)>0$ can I show $\mathbb{E}[\frac{T'x}{T'T}] > 0$?

x,T are vectors with $cov(x_i,T_i)>0$. Without specifying f(x,T), is it possible to determine the sign of $\mathbb{E}[\frac{T'x}{T'T}]$?
LucasMation's user avatar
1 vote
0 answers
68 views

Expected value of product of 2 correlated random vectors

Let x and y be complex Gaussian random vectors with length L, and $\mathbb{E}[x]=a$ and $\mathbb{E}[y]=b$, and they are correlated such that $c = \mathbb{E}\left[\left|\mathbf{\mathit{\mathbf{x}}}^{T}...
Shumpei's user avatar
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0 answers
709 views

(Co)Variance of a random matrix

The expected value $\mathbb{E}[\mathbf{x}]$ of a random vector $\mathbf{x} \in \mathbb{R}^{n \times 1}$ is the vector of the expected values of each individual random variable $\mathbf{x}$ contains. ...
user3389669's user avatar
0 votes
0 answers
25 views

probability density of a random vector greater than some value? [duplicate]

In single dimension, the probability that a random variable $X$ is greater than some value $x$ is easily related to the cumulative distribution(c.d.f.) as $Pr(X > x) = 1 - F(x)$ if only $Pr[X \leq ...
randomprime's user avatar
1 vote
0 answers
221 views

The joint distribution of Y=AX and Z=BX given a projection matrix A and residual maker matrix B, and a random vector X with known pdf?

This question follows on from a previous question I asked which was answered. It turns out my question lacked some important details, which was revealed by the answer posted on that thread. This is ...
h2learn's user avatar
  • 53
4 votes
1 answer
5k views

Distribution of the $L^{2}$ norm of a vector of components drawn from Gaussian distributions

I recently asked this question involving uniform distributions. I am wondering what would be the equivalent for Gaussian distributions. The problem states as follows. We consider a random vector $\...
Vincent's user avatar
  • 235
3 votes
1 answer
2k views

$L_2$ norm of product of two vectors

Let's assume we have two matrices $A^{d\times 1}$ and $B^{1 \times e}$, and we define their product as $C^{d\times e}$. Assuming $A,B$ are real valued with all entries in $[-1,1]$. I can intuitively ...
lvdp's user avatar
  • 387
0 votes
1 answer
36 views

Test for equality of means for vector-valued random process with different variances

I am studying linearity range of an RF amplifier (henceforth DUT). For that I am stimulating the DUT with a periodic deterministic probe signal (which is known only roughly) and measuring the DUT's ...
skobls's user avatar
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13 votes
3 answers
4k views

Does mean centering reduce covariance?

Assuming I have two non-independent random variables and I want to reduce covariance between them as much as possible without loosing too much "signal", does mean centering help? I read somewhere that ...
lvdp's user avatar
  • 387
7 votes
0 answers
1k views

Distribution of the $L^2$ norm of a vector of components drawn from uniform distributions

We consider a random vector $\vec{v} = \left(x_{1}, x_{2}, \dots, x_{n}\right)$ built from $n$ real random variables drawn from a real continuous uniform distribution $\mathcal{U\left(a, b\right)}$, $...
Vincent's user avatar
  • 235
0 votes
1 answer
492 views

Covariance of sums of pairs of correlated variables

Take two vectors of normally-distributed random variables $\mathbf{x} = (x_1, x_2, \ldots x_n)$ $\mathbf{y} = (y_1, y_2, \ldots y_n)$ where the covariance of each pair $(x_i, y_i)$ is known, $\...
Fab von Bellingshausen's user avatar
7 votes
2 answers
1k views

Suppose $\mathbf{X, Y}$ are independent random vectors. Are their components independent? [duplicate]

Let $\mathbf{X} = (X_1, \dots, X_p)^\top$ and $\mathbf{Y} = (Y_1, \dots, Y_p)^\top$ be independent. Does it then follow that $X_i$ is independent with $Y_j$ i.e. cov$(X_i, Y_j) = 0$?
Abdul Miah's user avatar
3 votes
1 answer
889 views

Conditional expectation of a vector

Suppose we have two random vectors $X=(X_1,X_2)^T$ and $Y=(Y_1,\dots,Y_n)^T$. I wish to find a simple definition or formula for $$ E_{X|Y=y}[X] $$ Intuitively, I think the following is correct: $$ ...
waynemystir's user avatar
1 vote
0 answers
70 views

Comparison of random vectors

I feel a little stupid asking this, but anyway: say I've got a random vector $a$ of size $p$ with covariance matrix ${\Sigma_a}$ of size $p\times p$, and another vector $b$ with its cov $\Sigma_b$. ...
Spätzle's user avatar
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