Questions tagged [random-vector]

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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 $\...
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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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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 ...
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1 vote
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28 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*} ...
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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. ...
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2 votes
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34 views

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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1 answer
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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 ...
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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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53 views

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 ...
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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 ...
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Covariance of the sum of two random vectors

This is the situation. I have an estimation of the position $(x_t,y_t)$ of an object with its covariance $\Sigma_p$ and an estimation of its speed $(v_x, v_y)$ with its covariance $\Sigma_v$. Actually,...
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1 answer
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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 ...
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1 answer
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$E[X^T (Y-Z)] = E[X^T] E[Y-Z]$ 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 ...
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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]) \...
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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 $$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$ (...
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1 answer
39 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 $\...
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2 answers
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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 ...
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7 votes
2 answers
437 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 $...
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4 votes
1 answer
175 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 ...
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0 votes
1 answer
100 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 ...
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239 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_{...
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1 vote
1 answer
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 $\mathbb{E}[\frac{T'x}{T'T}]$?
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1 vote
0 answers
46 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}...
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442 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. ...
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23 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 ...
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1 vote
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116 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 ...
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3 votes
1 answer
3k 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 $\...
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3 votes
1 answer
955 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 ...
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0 votes
1 answer
35 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 ...
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12 votes
3 answers
3k 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 ...
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5 votes
0 answers
924 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 $\mathcal{U\left(a, b\right)}$, $...
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  • 205
0 votes
1 answer
252 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, $\...
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5 votes
2 answers
609 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$(X_i, Y_j) = 0$?
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3 votes
1 answer
431 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: $$ ...
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1 vote
0 answers
51 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 $\Sigma_b$. ...
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