Questions tagged [covariance]

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

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44 views

The covariance between sum of random variables and maximum of random variables

Let $X_1,\ldots,X_n,\; n\ge1$ — independent random variables $U(0,1),$ $S_n=\sum_{i=1}^n X_i,$ $Z_n=\max(X_1,\ldots,X_n).$ Calculate $\mathrm{cov}(S_n,Z_n).$ Solution: $\mathrm{cov}(S_n,Z_n)=\...
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40 views

Mercer decomposition of transformed covariance kernel

I'm currently trying to determine the spectral representation of covariance kernels with the structure: $$K(s,t) = C \cdot [F(\min\{s,t\}) - F(s)F(t)], \quad s,t \in \mathbb{R}$$ of centered and ...
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27 views

Building covariance matrix from paths in the inverse covariance

The inverse of the covariance matrix for a multivariate normal random vector can be thought of as holding some measure of conditional dependence between two of the variables in the vector. Consider ...
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33 views

Find the variance of Y

In an old probability test: $$ P(X_i=1)=n^{-\frac{1}{2}} $$ $$ P(X_i=0)=1-n^{-\frac{1}{2}} $$ $$ S_{i,j,k} = 1 \text{ if }X_i=X_j=X_k=1 \text{ (and 0 otherwise)}$$ $$ Y = \text{Number of }S_{i,j,k}\...
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110 views

Change order of eigenvalues and correspoding eigenvector

I'm struggling with some bugs in my program and try to find the mistakes. The situation is as follows: I have a covariance Matrix on which I perform a principal component decomposition into a ...
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73 views

Do we really need finite variances $\mathrm{Var}(X)$ and $\mathrm{Var}(Y)$ in the definition of covariance $\mathrm{Cov}(X,Y)$?

You know that covariance of jointly distributed random variables $X$ and $Y$ is $$\mathrm{Cov}(X,Y) = \mathrm{E}[XY] - \mathrm{E}[X] \mathrm{E}[Y].$$ It is clear that we should require finiteness of $...
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119 views

Recovering the covariance matrix of pixel coordinate from normalised camera coordinate

lets assume we have the pixel value of some interest points as well as their covariance matrices that defined as follows: $$ \mathbf x^{\prime} = \begin{bmatrix} x^{\prime}\\ y^...
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32 views

Correlation between number of success (X) to total number of trials (N) [closed]

I have to come up with an expression that provides correlation between number of success( with probability p ) to total number of trials. The distribution of experiments is binomial. According to ...
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120 views

Covariance of independent random variables.

I've been fighting with following problem: Problem: Let $N, X_1, X_2, \ldots $ be independent random variables with given $\Lambda = \lambda$. Variables $X_i, i=1,2,\ldots$ have exponential ...
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70 views

Is this matrix singular and if yes, why?

Consider the following matrix \begin{align} \begin{pmatrix} p_1(1-p_1) & -p_1p_2 & \cdots & -p_1p_k \\ \vdots & \vdots & \ddots & \vdots \\ -p_kp_1 & -p_kp_2 & \cdots &...
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60 views

Is $X_t Y_t$ stationary?

If $X_t$ and $Y_t$ are independent and both are second order stationary processes, is $X_tY_t$ also stationary? I need to show that i) $E(X_tY_t)$ is time independent ii) $Var(X_tY_t)<\infty$ ...
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20 views

Joint optimization of precision matrices for common sparsity pattern

This question is motivated from paper by Cai, 2016 on joint estimation of multiple (K) precision matrices from K datasets. Let $X^{(k)} \sim N(\mu^{(k)}, \Sigma^{(k)})$ be a p-dimensional random ...
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55 views

Make a covariance matrix have a larger eigenvalue for one eigenvector

Suppose I have a covariance matrix $\Sigma$, with eigenvalues $u_1,u_2,u_3$ and eigenvectors $v_1,v_2,v_3$. How can I use that information to generate a new covariance matrix $\Sigma_2$ with the same ...
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58 views

Norm of covariance between scalar and vector

I would like to show that if $Y\in\Re$ is a random real vector such that $\|Y\|≤ \bar Y$ and $X \in \mathcal H$ where $\mathcal H$ is a Hilbert space (whose field is $\Re$), then $$ \|\operatorname{...
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1answer
371 views

Bounds on Eigen Values of a Covariance Matrix of Bounded Random Vectors

Suppose we have a random vector ${\bf X} \in \mathbb{R}^n$ where every element of ${\bf X}$ has a per element bound $ |X_i| \le a_i$. Now let ${\bf K}_{\bf X}$ be the covariance matrix of ${\bf X}$. ...
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1answer
503 views

Covariance of Brownian bridge increments

$\text{I need to prove the following:}\\$ $$\text{Cov}[\left(W(t_{i+1})-W(t_i)\right),\left(W(t_{j+1})-W(t_j)\right)|Z]= \begin{cases} \left(t_{i+1}-t_i\right)-\frac{\displaystyle\left(t_{i+1}-t_i\...
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176 views

Covariance matrix as linear transformation [closed]

Why applying the covariance matrix to the original vector will turn the vector to the direction of largest variance? How do I intuitively understand this? And what does the inverse of the ...
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311 views

Conditional PDF on Gaussian random vectors

Suppose the Gaussian random vector $\mathbf{X}\sim\mathcal{N}(\mathbf{\mu_X},\Sigma_\mathbf{X})$ where $$\mathbf{\mu_X}=\begin{bmatrix}1\\5\\2\end{bmatrix}$$ and $$\Sigma_\mathbf{X}=\begin{bmatrix}1&...
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1answer
81 views

Covariance of number of 1's and the sum of all results of $n$ fair die tosses

A die is tossed $n$ times. Find the covariance of the number of one's and the sum of all results. I started by defining a random variable $X_i$ as $ i=1,\ldots,n$, $X_i=1$ if at the $i$'th toss ...
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1answer
113 views

Covariance Matrix of Zero Mean Data

Given a covariance matrix is there a way to tell if it came from zero mean data?
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25 views

Finding subset of uncorrelated variables

Assume I have $n$ random variables with covariance matrix $\Sigma$. Now, I want to find $m$ groups of variables such that they very correlated inside each group, but their correlation between groups ...
2
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1answer
570 views

Why is the inverse of the sample covariance matrix a biased estimator of the true precision matrix?

So this paper makes a claim that sample covariance $S$ is an unbiased estimator of the true covariance $\Sigma$ - makes sense. However, if we make the inverse of said matrices $S^{-1}$ is no longer ...
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60 views

When can I be sure that the state values estimated from the Kalman Filter have approached the actual values? Is it from the state co-variance matrix?

Below are the equations for state estimation using Kalman Filter. Here are the first few equations, and the rest follows in a link below: $$ \newcommand{\blue}{\color{blue}} \newcommand{\grey}{\color{...
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1answer
39 views

A Covariance Problem! [closed]

I faced this problem in my master degree test and unfortunately I can't solve it :( The problem is: Lets $(X,Y)$ be a vector of random variables with probability distribution function: $$P(X=x,Y=y)...
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75 views

Simplify variance expression, taking into account covariances:$\operatorname{var}(Y_i-\bar Y-\hat {\beta}_1(x_i-\bar x))$

Find $\operatorname{var}(Y_i-\bar Y-\hat {\beta}_1(x_i-\bar x))$ where $\hat {\beta}_1=S_{xy}/S_{xx}$ is the least square estimator and $Y_i$ a random variable. I know that I can't simply split the ...
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680 views

Variance of a sum of identically distributed random variables that are not independent

I am "new" to probability/statistics and I was hoping someone could verify that this is correct. Let $Y_1,\ldots,Y_n$ be random variables that follow a common distribution with mean $\mu$ and variance ...
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391 views

Stationary Gaussian process whose correlation parameter approaches zero.

Consider a mean-zero ($\mu = 0$), unit-variance ($\sigma^2$) Gaussian random process $X(t)$. This process is strictly stationary (the joint-probability distribution does not vary with $t$). The ...
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44 views

What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
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88 views

Means and Covariances of powers of a normal distribution

Let $X$ be a normally distributed random variable, with mean $\mu$ and variance $\sigma^2$. Consider a random vector $$V = \left[ X^n, X^{n-1}, \dots, X^2, X, 1 \right]^T $$ What is the expected ...
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2answers
756 views

Covariance of Martingales

I have proven that Martingales have orthogonal increments. From this I need to show that $\operatorname{Cov}[M(t),M(s)]$ relies only on $\min\{s,t\}$. I used the expected value definition of ...
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1answer
43 views

Auto-covariance of Multiply Process

Given the process $I=XY$, where $X$ and $Y$ are: independent WSS with auto-covariances $B_X(\tau)=E\left\lbrace x(t)x(t+\tau)\right\rbrace-E^2\left\lbrace X\right\rbrace$ and $B_Y(\tau)$ $E\left\...
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105 views

Basic MVUE Application

I am having some trouble with the following problem: Let $X = (X_{1}, . . . , X_{n})$ a random sample from $f_{\theta}$, where $\theta \in \Theta$. Suppose that $W$ is the MVUE for $\theta$. Let $Z$ ...
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1answer
460 views

Quaternion Kalman Filter Process Noise

I'm implementing a extended Kalman filter using quaternions. I've extended this paper to deal with my custom observations. My state space is analogous to the one in the previous paper : $ \mathbf{...
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512 views

Expected value of the sample covariance

Let $X = (X_1,\dots,X_p)$ is a random (column) vector with values in $\mathbb R^p$. The covariance matrix $\mathrm{Cov}(X,X)$ is defined by $$\mathrm{Cov}(X) := E[(X-E[X])(X-E[X])^T]$$ By definition ...
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162 views

If a Stochastic Process has Variance linear with t, how to prove it is not Wide Sense Stationary?

For my study, as a part of a Matlab exercise, the following question is asked: Using the results of the estimated standard deviations of the random variable $x(k)$ for $k = 10^3; 10^4; 10^5$ ...
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2answers
216 views

zero covariance but not independent - normally distributed random variable $X$ and $X^2$

This is one of my homework question, which the answer sheet has already been given out. However, I still don't understand it. Exercise 1.1. It is well known that for two normal random variables, zero ...
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1answer
20 views

linear modeling question: How can I find variance for vector Y?

Linear models are $$Y_1= 2\theta_1+3\theta_2 +\epsilon_1$$ $$Y_2= -2\theta_1+\theta_2 +\epsilon_2$$ and $\epsilon_1=3z_1-z_2$ and $\epsilon_2=4z_1+z_2$, where $z_1, z_2$ are two random variance such ...
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66 views

Variance and covariance

I'm practicing for an exam and a mock question has me completely stumped. If someone could show me the steps I would be very grateful! There are two random variables, $A$ and $B$. $Var(A) = 9$, and $...
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1answer
974 views

Expectation of the product of two dependent binomial distributions.

Let $X_i$ (i ∈ N) be independent and identically distributed all following a Bin(10, p) distribution for some value p ∈ [0, 1]. Define $Y_n$ := $$\sum_i^m X_i$$ Compute $t_n,m$ = Cov($Y_n$,$Y_m$) for ...
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483 views

Singular covariance matrix, understanding the beginning of a proof

Let $\left\{X_t,t\in T\right\}$ be a stationary process such that $\text{Var}(X_t)<\infty$ for each $t\in T$. The autocovariance function $\gamma_X(\cdot)=\gamma(\cdot)$ of $\left\{X_t\right\}$ is ...
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2answers
1k views

What is meant by statisticians when they talk about between population differences vs within population differences?

Suppose we have two populations of people in different parts of the world and we want to talk about the variation in heights between the two populations. As I understand it, statisticians are talking ...
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19 views

Find $W\in\mathbb{R}^{T \times N}$ such that for $X\in\mathbb{R}^{T \times N}$: $X'X=W'W-\frac{1}{T}W'\iota_T\iota_T'W$

Given $X\in\mathbb{R}^{T \times N}$, I would like to know how to find $W\in\mathbb{R}^{T \times N}$ such that $X'X=W'W-\frac{1}{T}W'\iota_T\iota_T'W$. $\iota_T$ denotes the vector of ones of length T. ...
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0answers
110 views

Distribution of unknown covariance matrix, given variance of linear combination

Suppose I am uncertain about the covariance of a vector-valued random variable $X$, but the variance of some linear combination is known. How does this affect the distribution of $X$? Specifically $X$...
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1answer
26 views

Problem with verifying variance of residual

I am supposed to show the following: $$Var(e_{ij}) = \sigma^{2}\left(1-\frac{1}{n_i}\right)$$ Where the follwing is known: $$y_{ij} = \mu + \alpha_{i} + \varepsilon_{ij}$$ $$e_{ij} = y_{ij} - \hat{...
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267 views

Covariance of absolute value of variables

I have some relatively complicated variables I am working with. I need to find the covariance: $\text{cov}(\left|x\right|,\left|y\right|)$, is there a simplification of $\text{cov}(\left|x\right|,\...
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1answer
160 views

centering two variables $X$ and $Z$ makes $cov (X,XZ) = 0$

I've read that centering two normal (or symmetrical) variables $X$ and $Z$ affects correlation of centered $X$ with interaction term $X\cdot Z$ in such way, that this correlation $cor(X-EX, X\cdot Z)$ ...
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0answers
3k views

Standard Error of Sample Variance

I have a time-series of values $X_1, X_2, \ldots, X_t$, for which I compute sample variance: $$\hat{\sigma}^2 = \operatorname{var}(X_1, \ldots, X_t)$$ (unabiased estimator using $\frac{1}{t-1})$. ...
2
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1answer
50 views

Covariance of a mixture of Gaussians

I have seen this question asked, but in a strange way that I do not think is equivalent. If someone can show that the formulations are identical, I would be grateful. Suppose with probability $p$, ...
2
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2answers
85 views

uniqueness of joint probability mass function given the marginals and the covariance

Let X and Y be two nonnegative, integer-valued random variables. Is there a way to find the joint probability mass function, i.e. $$ \mathbb{P}(X= k, Y= h) $$ for some $k,h\geq 0$, given the ...
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0answers
44 views

Development of a covariance

I have to compute a covariance. But I have some difficulties. My covariance has the following shape : $$ \sum\limits_{a=1}^t\sum\limits_{b=1}^t\sum\limits_{c=1}^t\sum\limits_{d=1}^t\operatorname{cov}\...