Questions tagged [random-vector]
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35
questions
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vote
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17
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 $\...
3
votes
1
answer
121
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 ...
1
vote
0
answers
54
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 ...
1
vote
0
answers
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*}
...
0
votes
1
answer
56
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.
...
2
votes
0
answers
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 ...
0
votes
1
answer
6
views
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 ...
0
votes
0
answers
21
views
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 $j \times m$...
1
vote
0
answers
53
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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 ...
3
votes
1
answer
91
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 ...
0
votes
0
answers
37
views
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,...
0
votes
1
answer
85
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 ...
1
vote
1
answer
53
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 ...
7
votes
1
answer
105
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:
$$
\begin{align}
R_X &= ([0,1] \times [0,1]) \...
1
vote
0
answers
33
views
What is the entropy of multivariate data multiplied by a vector?
It is a general rule that for multivariate data $\boldsymbol{X}$ and a matrix $\boldsymbol{A}$, their entropy is
$$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$
(...
0
votes
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 $\...
0
votes
2
answers
92
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 ...
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 $...
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 ...
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 ...
0
votes
0
answers
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_{...
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}]$?
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}...
0
votes
0
answers
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.
...
0
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0
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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 ...
1
vote
0
answers
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 ...
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 $\...
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 ...
0
votes
1
answer
35
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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 ...
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 ...
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)}$, $...
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,
$\...
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$?
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]
$$
Intuitively, I think the following is correct:
$$
...
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 $\Sigma_b$.
...