Questions tagged [random-vector]
The random-vector tag has no usage guidance.
38
questions
1
vote
0
answers
42
views
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 ...
- 59
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 ...
- 21
1
vote
0
answers
116
views
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 ...
- 31
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 ...
- 2,385
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 $\...
- 61
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 ...
- 767
1
vote
0
answers
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 ...
- 133
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*}
...
- 208
0
votes
1
answer
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.
...
2
votes
0
answers
53
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 ...
user339651
0
votes
1
answer
10
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 ...
- 255
0
votes
0
answers
42
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 $j \times m$...
- 571
1
vote
0
answers
87
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 ...
- 419
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 ...
- 31
0
votes
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 ...
- 11
1
vote
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 ...
- 765
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:
$$
\begin{align}
R_X &= ([0,1] \times [0,1]) \...
- 384
0
votes
0
answers
82
views
What is the entropy of multivariate data multiplied by a vector?
It is a general rule that for multivariate data $\boldsymbol{X}$ and a matrix $\boldsymbol{A}$, their entropy is
$$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$
(...
- 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 $\...
- 113
0
votes
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 ...
- 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 $...
- 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 ...
- 53
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 ...
- 31
0
votes
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_{...
- 1,313
1
vote
1
answer
36
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}]$?
- 71
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}...
- 11
0
votes
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.
...
- 243
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 ...
- 365
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 ...
- 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 $\...
- 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 ...
- 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 ...
- 1
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 ...
- 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)}$, $...
- 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,
$\...
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$?
- 153
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]
$$
Intuitively, I think the following is correct:
$$
...
- 145
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 $\Sigma_b$.
...
- 3,400