# Questions tagged [covariance]

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

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### Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance

I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement: ...
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### What does Determinant of Covariance Matrix give?

I am representing my 3d data in convariance matrix. I just want to know what the determinant of Convariance Matrix gives. If the determinant is positive, zero, negative, high positive, high negative. ...
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### When does the inverse of a covariance matrix exist?

We know that a square matrix is a covariance matrix of some random vector if and only if it is symmetric and positive semi-definite (see Covariance matrix). We also know that every symmetric positive ...
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### Prove that $U=Y - E[Y|X]$ and $X$ are uncorrelated

Let $U = Y - E[Y|X]$. How can I prove that $U$ and $X$ are not correlated? I've been doing a lot of things but when I calculate $\text{cov}(U,X)$ I finish with $EXY - EXEY$ and not $0$ which would be ...
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### Variance of Z = max(X,Y) where X Y are jointly bivariate normal

I have a question about the bivariate normal r.v.'s Given $X, Y \sim \operatorname{Normal}(0,1)$ with correlation coefficient $\rho$. Let $Z=\max(X,Y)$. Show that $\operatorname E Z^2=1$. My ...
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### Covariance of $X^2$ and $X^3$ when $X$ is exponentially distributed

Here is my work.... \begin{align*} Cov(Y,Z) &= E(YZ) - E(Y)E(Z)\\ &= E(X^2\cdot X^3) - E(X^2)E(X^3)\\ &= E(X^5) - E(X^2)E(X^3) \end{align*} And we know $E(X^n) = \frac{n!}{\lambda^n}$...
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### Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
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### 3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix

This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ...
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### Prove that Cov(X,Y)=Cov(X,E[Y|X])

I've been working on this problem for 3 hours now, and my complete lack of progress is getting disheartening. I've looked up definitions, proofs, and have even seen a solution for this particular ...
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### A question about a generalization of covariance

Suppose, $H$ is a Hilbert space over $\mathbb{R}$. Suppose, $X$ and $Y$ are random vectors in $H$. Let’s define Hilbert expectation of a random vector $X$ in a Hilbert space $H$ as a vector $v \in H$, ...
Suppose that $X$ and $Y$ are positive and square-integrable random variables, such that $X$ and $Y$ are positively correlated, i.e., $\mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] \geq 0$. Let $f: \... 1answer 166 views ### Given a set of 3 orthogonal vectors, how can I find a minimum volume enclosing ellipsoid expressed in the Cartesian coordinate frame? Generalized Problem Given values to start the problem: A 3D orthonormal coordinate frame (we'll call it the 'V' coordinate frame) that is rotated from the global coordinate system (we'll call it 'G') ... 2answers 451 views ### Why does the variance formula has a square term? I was reading about variance from Head First Statistics : And then - Q. I find the reasoning a little absurd. Wouldn't just taking the absolute distance suffice if cancelling out of the terms was ... 0answers 201 views ### computing the inversion of a matrix which is the sum of a Kronecker product and an identity matrix I'd like to evaluate a single entry$s_{ik}$of the$\mathbf{S}$matrix, using Markov chain Monte Carlo approach. The posterior of$\mathbf{S}$has a Gaussian likelihood with a covariance matrix $$\... 1answer 86 views ### Finding two tangential vectorfields with Lie-Bracket equal zero I've been dealing with the following problem for a while and meanwhile I have no idea how to move on. Maybe one of you can help me? :) Let M = \mathbb{R}^2\subset \mathbb{R}^3 a manifold. I need ... 0answers 163 views ### Applying PCA on covariance matrix in order to generate a new random variable. Let \mathbf{x} be a random n\times1 real vector, \mathbf{x}\in\Bbb{R}^n, which is distributed normally with mean \bar{\mathbf{x}} and covariance matrix \Sigma_x\in\Bbb{R}^{n\times n}, i.e. \... 0answers 349 views ### What is {\rm cov}(e_i, \hat y_i) in simple linear regression? The model is y_i = \beta_0 + \beta_1x_i + \epsilon_i What is {\rm cov}(e_i, \hat y_i)? What is {\rm cov}(\epsilon_i, \hat \beta_1)? What is {\rm cov}(e_i, \epsilon_i)? For 1, I am writing ... 1answer 89 views ### Is this statement true: \mathsf{Cov}(X,Y)\geq 0, \mathsf{P}(Y>0)=1, show that \mathsf{Cov}\Big(X,\dfrac{1}{Y}\Big)\leq 0 I was trying to prove a problem and I got stuck at a point. The problem leads to a point where I have to show: Let, X and Y are two r.v. If \mathsf{Cov}(X,Y)\geq 0 and \mathsf{P}(Y>0)=1, ... 1answer 493 views ### Cholesky decomposition and variance Well, I've been reading about simulating correlated data and I've come across Cholesky decomposition. Everything seemed clear until I found a couple of posts on this site and Cross-Validated that ... 1answer 49 views ### Understanding Variance-Covariance Matrix Suppose data set is expressed by the matrix X \in\mathbb R^{n \times d} where n = Number of samples and d = dimension/features of each sample Then what does \operatorname{Cov}(X) \in\mathbb R^... 1answer 41 views ### Random variables and co-variance, Statistics 318 For the given example in the book John E. Freund's Mathematical Statistics with Applications, 8th edition, by Miller and Miller. ISBN: 9780321807090 I've highlighted using colors what numbers ... 2answers 721 views ### Covariance matrix of uniform distribution over the Sierpinski triangle Let (X_1, X_2) be uniform over the unit Sierpinski triangle (represented in Cartesian coordinates). What is its covariance matrix? This is a question I saw in a jobs ad. I would love some leads on ... 1answer 3k views ### Covariance Matrix of mean-centered Random Variables I read here that for n x d data matrix X, where X is mean-centered, V = X^{T}*X is its covariance matrix. Why is that? As I understand the element V_{i,j} of the covariance matrix is defined by ... 3answers 236 views ### \operatorname{Cov} \, (A,B)\geq 0 and \operatorname{Cov}(B,C)\geq 0\Rightarrow \operatorname{Cov}(A,C)\geq 0? Let's say we have three random variables A, B and C. I know that \DeclareMathOperator{cov}{Cov} \cov(A,B)\geq 0 and \cov(B,C)\geq 0 . Then is it true that \cov(A,C)\geq 0? 1answer 195 views ### efficient way to invert a Matrix plus a diagonal one Let \Sigma be a n \times n matrix, V a 2 \times 2 matrix and I_{2 n} the identity matrix on dimension 2n \times 2n. Both \Sigma and V are covariance matrices, thus real, symmetric and ... 2answers 76 views ### Show, that \frac {E[Xg(X)]}{E[g(X)]} \ge E[X] when g strictly monotonic increasing Let g:\mathbb R \to (0,\infty), X real valued random variable and g(X) \in \mathcal L^2 and g strictly monotonic increasing. Show, that \frac {E[Xg(X)]}{E[g(X)]} \ge E[X] I tried something ... 1answer 213 views ### How to prove that \mathrm{var}(X-E(X|Y)) \leq \mathrm{var}(X)? I tried to solve this exercise but got stuck: Assume we have the random variables X and Y where E(X) = 0. How can we prove the following inequality \operatorname{Var} (X-E(X|Y)) \leq\... 1answer 2k views ### What's the best way to think about the covariance matrix? Let X be a random vector with covariance matrix \Sigma. People often describe \Sigma in terms of its components: \Sigma_{ij} is the covariance of the ith and jth components of X. But ... 1answer 130 views ### perfectly correlated processes I am really stuck in this question: Let \{S_t\} and \{S'_t\} be two stochastic processes, satisfying \begin{equation} dS_t = S_t ( \sigma_t \,dB_t + r_t \,dt), \quad dS'_t = S'_t (\sigma'_t \,... 1answer 104 views ### Covariance of two expectation estimators that used different numbers of samples Say I have two estimates of the mean of two functions:$$Q^1_{N_1}=\frac{1}{N_1}\sum_{i=1}^{N_1}f^1(X_i), \quad Q^2_{N_2}=\frac{1}{N_2}\sum_{i=1}^{N_2}f^2(X_i),$$where each sample$X_i$is identical ... 1answer 268 views ### Why do polynomial regressions have larger variance at the end? In reading the book "An Introduction to Statistical Learning with Applications in R", I came across this graph: It shows that the point-wise variance is larger at the ends of the regression curve. ... 1answer 156 views ### Which matrices are covariances matrices? Let$V$be a matrix. What conditions should we require so that we can find a random vector$X = (X_1, \dots, X_n)$so that$V = Var(X)$? Of course necessary conditions are: All the elements on ... 0answers 104 views ### The variance of a covariance The expectation over$x$of a covariance between variables$A$and$B$(where the distribution of$B$varies according to$x$) is equal to the covariance of$A$with the expectation of$B$over$x$:$...
If we say $\operatorname{Var}(x)$ has $n-1$ degrees of freedom which are lost after we estimate $\operatorname{Var}(x)$, this matches how $n-1$ observations are now constrained to be sufficiently ...