Questions tagged [random-vector]
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13
questions
0
votes
0answers
21 views
Covariance matrix as a sum of two covariance matrices
Suppose that a random vector $\mathbf{n}$,
$$\mathbf{n} = \mathbf{n}_A + \mathbf{n}_B \ , \tag{1}$$
can be written as a sum of two random vectors $\mathbf{n}_A$ and $\mathbf{n}_B$, that are ...
1
vote
0answers
20 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
0answers
26 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 ...
0
votes
1answer
42 views
How can I prove the following relation between the probabiloity of X and its expectation using Cauchy-Schwarz inequality?
For a random variable $ X \geq 0 $ and $E[X^2] < \infty $, I'm asked to prove the following:
$ P(X> 0) \geq \frac{(E[X])^2}{E[X^2]}$
It makes intuitive sense to me that it must be the case, ...
3
votes
1answer
182 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
57 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
1answer
25 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 ...
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 ...
2
votes
0answers
124 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
1answer
18 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,
$\...
4
votes
2answers
113 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$(X_i, Y_j) = 0$?
1
vote
1answer
51 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:
$$
...
0
votes
0answers
39 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 $\Sigma_b$.
...
