Questions tagged [covariance]

Questions about covariance, a measure of (linear) association between two random variables.

Filter by
Sorted by
Tagged with
0
votes
1answer
17 views

Joint distribution and covariance of Poisson process and waiting time

Hi I am having a trouble solving for this problem where I have to find 1) Joint distribution of $W_{1}$, $W_{r}$ for $r\geq2$. 2) $\operatorname{Cov}(W_{1},W_{r})$ for $r\geq2$. [Notation ...
0
votes
1answer
34 views

Covariance of $X$ and $Y^2$ where $(X,Y)$ is bivariate normal

I'm trying to solve a case where there is bivariate random vector $(X,Y)$ that has the bivariate normal distribution below ($-1<\rho<1$): $$\begin{pmatrix} X\\ Y \end{pmatrix}\sim N_{2}\...
-1
votes
1answer
30 views

product of normally distributed random variable with discrete random variable [closed]

I have to solve the following problem, which doesn't look difficult to me but I can't seem to solve it: Let $N$ be a real valued Gaussian random variable with mean zero and variance $1$ and let $Z$ ...
0
votes
1answer
31 views

When is $\sqrt{k}\operatorname{Var}\left(\sum_{i=1}^{k}X_{i}\right) \leq \sum_{i=1}^{k}\operatorname{Var}(X_{i}) $ true?

Assume we have $k$ dependent random variables $X_{1}, \dots, X_{k}$ with $\operatorname{Var}(X_{i}) < \infty$. In which case $$\sqrt{k}\operatorname{Var}\left(\sum_{i=1}^{k}X_{i}\right) \leq \...
0
votes
0answers
20 views

Covariance of squared terms

Let (X,Y) both follow weibull distribution and cov(X,Y)=c. Determine the cov(X^2,Y^2). I can't make it out. Any help is appreciated.
0
votes
0answers
24 views

Calculate covariance of 2 random variables when only given variance and cov

So I've been given the following exercise: Let $X$ and $Y$ be two random variables. Let $Var(X) = 2$ and $Cov(X, Y) = 1$. Compute $Cov(5X, 2X + 3Y)$. How do I do this when I don't have the second ...
0
votes
0answers
22 views

find the correlation coefficient of two random variables after the same function transformation

we have two random variables: U1 and U2 follow uniform distribution between 0 and 1: U1 ~ U(0,1), U2 ~ U(0,1) and correlation: corr(U1,U2) = ρ covariance : cov(U1,U2) = corr(U1,U2)/12 Then we do ...
0
votes
1answer
11 views

Markov chain covariance calculation

Suppose $Y_t=1$ with probability $\pi_s$ and $0$ with probability $1-\pi_s$. Moreover, we know that $Y_t$ is a Markov chain with transition probabilities $Pr(Y_t=1|Y_{t-1}=1)=(p^3+(1-p)^3)/p_s$ and $...
0
votes
0answers
28 views

Covariance of two cdfs of two standard normal distribution [duplicate]

Let X1 and X2 be standard bi-gaussian distribution with correlation ρ: X1~N(0,1), X2~N(0,1), cov(X1,X2) = ρ Then Y1 and Y2 are cdfs of X1 and X2, we know they are random variables of uniform ...
2
votes
1answer
17 views

Covariant and contravariant components of vectors

I am struggling with the covariance and contravariance of vectors. In my physics classes, the professor explained that if covariant components transform with a certain matrix, then contravariant ...
0
votes
1answer
20 views

Covariance of a joint PDF with a min function.

I have to find the covariance of the following PDF $$f(x,y)=\begin{cases} 3\min\{x,y\}&, \text{ if }\ 0<x,y<1 \\ 0 &, \text{ otherwise} \end{cases}$$ Therefore I need to find $E(X)$ ...
0
votes
0answers
5 views

Decomposition of Gaussian spaces with respect to covariance function

Let $K(t,s):T^2 \to \mathbb{R}$ be a kernel symmetric and type positive (for every $n$ $\sum^n_{i,j}u_iu_jK(t_i,t_j) \geq 0$ and $(u_1,\dots,u_n) \in \mathbb{R}^n$) where $T$ is any set. Thus, it is ...
0
votes
0answers
19 views

Covariance with Sums of Random Variables

If $Q=10X_2,X_3.....X_n$ and $Z=X_1,3X_2....X_{n}$, then Is it also the case $Cov(Q,Z)=Cov(X,X)$=Var[X]? If this is the case, what is to be done with the coeifficient of 10 in front of $X_1$ in ...
0
votes
0answers
2 views

How can we find the covariance of sum of random variables (subgroup means) with their combined mean? [closed]

I need the solution (precise form) of $\text{Cov}(\sum_{i=1}^{t-1} \bar X_i ,\sum_{i=1}^{t-1} \bar{\bar X}_{i-1})$, where $\bar{\bar X}_0=\mu_0$(constant), $\ \bar{\bar X}_{i-1}=\sum_{j=1}^{i-1}\bar ...
0
votes
0answers
25 views

Covariance matrix problem

We consider 2 independent throws of the dice. We note the faces of the dice with $0,1$ and with $X,Y$ the random variables corresponding to the first and second throw. What is the covariance matrix ...
0
votes
0answers
21 views

Finding the inverse of a specific block matrix.

I have a problem where I have to calculate the inverse of a specific block matrix: $$ \begin{pmatrix}M^{-1}+B\Lambda^{-1}B^T & B\Lambda^{-1}A^T\\ A\Lambda^{-1} B^T & L^{-1} + A\Lambda^{-1}...
0
votes
1answer
49 views

Show covariance of random variable and an increasing function is increasing with respect to the mean

Suppose I have a continuous random variable $Y$ with $\mu=E[Y]$ and $g(Y)$ is strictly convex and increasing in $Y$. Does it follow that $\frac{\partial}{\partial\mu}Cov(Y,g(Y))>0$? To me, it ...
0
votes
0answers
16 views

Variance of draw from a multivariate normal distribution

I have a question that I cannot answer myself. I am close and seem to know the answer, but I do not know how to get there. My question is closely related to this question. That is, let's call the ...
0
votes
1answer
16 views

Recovering the covariance matrix from directional variance

How to find the covariance matrix $K$ (in some basis) of dimension $n$, given $\{$projected variance of $K$ along $V \enspace:\enspace \forall \enspace n$-dimensional vector $V\}$? In other words, if ...
0
votes
0answers
27 views

Simple question about calculating covariance with random variables

Consider two random variables $X,Y$ with $X$ having values in ${2,3}$ and $Y$ in ${4,7}$, each with probability $0.5$. This then means that both random variables are equally distributed. Calculating ...
0
votes
0answers
12 views

Evaluate Cov(A, BS)

Suppose I have three real-valued random variables A, B and S with poisson or negative binomial distribution. Let 𝑐𝑜𝑣(⋅) denote the covariance operator. How do I evaluate Cov(B, AS)? I have ...
1
vote
1answer
33 views

problem finding the Variance of dependent variables using covariance and correlation

Hello I have been trying to figure out this question for a few hours and I am very stuck and don't know how to progress but I am pretty sure I am wrong and would greatly appreciate some help. $X$~$N(\...
0
votes
0answers
37 views

What does the multiplication mean in this context?

I am trying to understand the intuition behind using multiplication, especially for the context of calculating things like the covariance, correlation, R-squared, etc. for example, I know that, in ...
0
votes
1answer
16 views

Calculating the variance of $\overline{X_N}$

Consider the sequence $(X_n)_{n \in \mathbb{N}}$ of random variables all with $\mu = 0$ and $\sigma^2 = 1$. Further, we have that $$\text{Cov}(X_i, X_j) = \begin{cases} 0 & \text{if} \: \: |i-j| &...
0
votes
0answers
9 views

Linear combination of multivariate sub-gaussian

Suppose $x \in \mathbb{R}^{m}$ and $y \in \mathbb{R}^n$ are independent zero-mean sub-gaussian random vectors both with covariance $I$. Let $A \in \mathbb{R}^{m \times m}$ and $B \in \mathbb{R}^{m \...
0
votes
0answers
13 views

Covariance generated from best-fit chi error function

I came across definition of covariance matrix that is defined from best fit error equation. I would like to clarify correctnes of procedure. We define equation for error: \begin{equation}\label{eq:...
0
votes
0answers
12 views

Which covariance matrix should I use for treating heteroskedasticity in my panel data?

I have a data set with panel structure (panel data) with 78 individuals observed over 5 three-year periods. I have 10 dependent variables an 1 independent variable. I applied logarithmic ...
0
votes
0answers
42 views

Connection between zero covariance of two random variables and their conditional expectation

I had an assignment to find covariance, conditional probability densities and conditional expectations of random variables $X$ and $Y$ with joint probability function $f(x,y) = 9x^2y^2$, $0 \le x \le ...
0
votes
1answer
20 views

Does this property of covariance require independence between random variables?

Lemma 5.3.6 in this page states that: $$\textrm{cov}(X+Y,Z)=\textrm{cov}(X,Z)+\textrm{cov}(Y,Z)$$ Does this property require that X, Y and Z must be pairwise independent?
0
votes
0answers
20 views

Covariance of $X$ and $M$

I was wondering how we compute the covariance of $r$$x$ and $r$$m$. Here is the problem: We know that: For stock $X: E(r_x) = 0.21$, and $Stdev(r_x) = 0.15$. For stock $Y: E(r_y) = 0.15$, and $...
0
votes
1answer
26 views

iid random samples

Let $X_1,X_2,\ldots,X_n $ be iid random samples from $N_p(0,Σ)$ and $b = \sum_{i=1}^nc_iX_i$ and $\sum_{i=1}^nc_i^2 = 1$ I am trying to show $b = \sum_{i=1}^nX_iX_i^T-bb^T$ is independent of $bb^T$? ...
0
votes
1answer
12 views

Covariant derivative formula- question about change of index

enter image description here How can you simply change superscript index j to k only for the 1st v & not for the 2nd v in the equation as shown in the diagram?
0
votes
1answer
34 views

Application of Markov Inequality(and/or Chebyshev's Inequality) to prove multivariable case

I'm trying to solve a problem that I think uses Markov Inequality(or chebyshev). But not sure how to apply it. the problem is like this: Show that $$P(|X-E(X)|+|Y-E(Y)|\geq4\sigma)\leq\frac{1}{2}....
1
vote
0answers
12 views

Generalizations of covariance to non-linear dependencies

I have been learning about statistics and probability theory superficially over last two-sih years, and on many occasions I have stumbled upon statements like: "The covariance gives some sense of how ...
0
votes
0answers
26 views

Computing the bias of the sample autocovariance with unknown mean

In "Introduction to statistical time series" by W. A. Fuller (1976), two definitions of the sample autocovariance with lag $h$ for a signal of length $n$ and unknown mean are proposed, namely $$\...
1
vote
0answers
26 views

Is there any way to estimate the covariance matrix of a nonlinear function?

Suppose $\mathbf{x}=f(\mathbf{y})$ where vectors $\mathbf{x}$ and $\mathbf{y}$ $\in R^N$, and the covariance matrix of $\mathbf{y}$ is given as $\sigma_y$. How do we estimate the covariance matrix of $...
0
votes
1answer
7 views

Covariance function of a Gaussian Process, sin and cos

Let $X(t) = \varepsilon_1 \cos(t)+ \varepsilon_2 \sin(t)$ where $\varepsilon_1, \varepsilon_2 ∼ N(0, 1)$ iid. Find the covariance function $C_X(s, t)$. I know that E(X) = 0 so Cov(X(s),X(t)) = E(X(s)...
2
votes
1answer
57 views

How to deal with underflow issues in high-dimensional entropy calculation?

I was not sure if the question makes sense here or should better be placed in a computational/CS forum, but I hope you can give me some insights. I am working in image processing and use the ...
2
votes
1answer
32 views

Computing Covariance of Sums of i.i.d. Random Variables

Problem: Suppose that $X_1,X_2,\dots$ are i.i.d. random variables with $E[X_1]=1$ and $E[X_1^2]=5$. Let $S_n=X_1+\cdots+X_n$. Compute $\text{Cov}(S_a,S_b)$ for $1\leq a<b.$ Attempt: By the ...
0
votes
0answers
10 views

What does $cov(x_1,x_2) >> 0, cov(y_1, y_2) >> 0$ and $cov(x_1+y_1, x_2+y_2) = 0$ tell us about $x_1, x_2, y_1, y_2$?

I was posed this problem where we know: $$ cov(x_1,x_2) >> 0 \\ cov(y_1, y_2) >> 0 \\ cov(x_1+y_1, x_2+y_2) = 0 \\ $$ What does this tell us about the structure of $x_1, x_2, y_1, y_2$? ...
0
votes
0answers
8 views

Usage of empirical auto-covariance matrices of time series

Let $x=(x_t)_{t\in \mathbb N}$ be a multivariate stochastic process that takes values $x_t\in\mathbb R^n$. Suppose we have a sample path of $x_t$ as $\bf X$, where $\mathbf X=(\mathbf x_1,\mathbf x_2,...
1
vote
1answer
25 views

Squared exponential kernel with Manhatten distance does not result in positive semi-definite matrix

Here it is stated that the squared exponential covariance function $$C(d) = e^{-(\frac{d}{V})^2},$$ where $V$ is a scaling parameter and $d$ is a distance between two points, is a stationary ...
0
votes
1answer
35 views

Multivariate normal distribution with one constant term

Assume that we have a random vector $X$ taking values from $\mathbb{R}^{k}$ which follows a multivariate normal distribution, i.e. $X \sim \mathcal{N}(\mu, \Sigma)$, where $\mu \in \mathbb{R}^{k}$ is ...
0
votes
2answers
47 views

If the Covariance between $(X,Y)$ is finite, then the second moment of $X$ is finite? [closed]

If the Covariance between $(X,Y)$ is finite, then the second moment of $X$ is finite. I think that this statement is false, but I can´t find a counterexample....
1
vote
0answers
15 views

Are two functions of the same random variable dependent? Is there a proof for this?

Suppose x is a random variable and f(x) and g(x) are functions. Is there a theorem that says f(x) and g(x) must be dependent so that cov(f(x),g(x)) is non-zero? The particular problem I'm dealing ...
0
votes
1answer
35 views

Coin Flip Covariance

You flip a coin 2n times. The probability the coin shows heads is $p$, where $0<p<1$. Let $X$ be the number of times you see heads in the first $n$ flips, and let $Y$ be the number of times you ...
0
votes
1answer
18 views

Covariance of colored cards

There is a 20 card deck that contains 12 red cards, 4 black cards and 4 blue cards. You draw (without replacement) five cards. Let X be the number of red cards drawn and Y be the number of black cards ...
0
votes
0answers
16 views

Find $\text{Cov}(N_{1}(t),N_{2}(t))$

Let $\{M_{i}(t), t \geq 0\}$, $i=1,2,3$ be independent Poisson processes with respect rates $\lambda_{i}$, $i=1,2,$ and set $$N_{1}(t)=M_{1}(t)+M_{2}(t), \quad N_{2}(t)=M_{2}(t)+M_{3}(t)$$ The ...
-1
votes
1answer
23 views

How to calculate the covariance of a matrix?

I can't justify which method is correct and why. If I have an $(M x N)$ matrix and I want to calculate its covariance, would mean center the matrix and then do $$ X^TX $$ or... $$ XX^T $$ does ...
1
vote
1answer
25 views

Making sense of covariance of space-time white noise as a product of delta distributions

The covariance of space-time white noise $\dot{W}(x,t)$ is given by $\mathbb{E}\dot{W}(x,t)\dot{W}(y,s) = \delta(t-s)\delta(x-y)$, where the $\delta$-distribution satisfies $\delta(x) = 0$ if $x\neq 0$...

1
2 3 4 5
28