Questions tagged [random-vector]

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11
votes
3answers
2k 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 ...
7
votes
1answer
90 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]) \...
6
votes
2answers
279 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 $...
5
votes
0answers
484 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 $\mathcal{U\left(a, b\right)}$, $...
4
votes
2answers
204 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$(X_i, Y_j) = 0$?
4
votes
1answer
67 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 ...
3
votes
1answer
1k 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 $\...
2
votes
1answer
339 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 ...
1
vote
1answer
124 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: $$ ...
1
vote
1answer
33 views

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 $\mathbb{E}[\frac{T'x}{T'T}]$?
1
vote
0answers
24 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 $$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$ (...
1
vote
0answers
41 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}...
1
vote
0answers
47 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 ...
1
vote
0answers
44 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 $\Sigma_b$. ...
0
votes
1answer
50 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 ...
0
votes
2answers
36 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 ...
0
votes
1answer
22 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 $\...
0
votes
0answers
18 views

Published source for D-dimensional behaviour of Dot-Product

I am currently studying the behaviour of the dot product between two random vectors in $R^d$. Specifically I wanted to start with the case of uniform random vectors on $\mathcal{S}^{d-1}$. I found ...
0
votes
0answers
8 views

Correct way to present the definition a of Markov process of order $p$ for a vector process?

Usually when we define a Markov process of order $p$ for a univariate time-series $\{X_t\in\mathbb{R},t=1,2,\cdots\}$, the definition is presented as follows \begin{equation} P(X_t\leq x_t\mid x_1,\...
0
votes
0answers
21 views

Distribution of the dot product between random unit vectors [duplicate]

Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases: $X,X'$ independent with uniform distribution on the sphere $...
0
votes
1answer
39 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 ...
0
votes
0answers
66 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_{...
0
votes
0answers
220 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. ...
0
votes
0answers
22 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 ...
0
votes
1answer
29 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 ...
0
votes
1answer
117 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, $\...