Questions tagged [covariance]

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

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2k views

Decorrelating variables using Cholesky decomposition

I am looking for a method to decorrelate several variables, so that their covariance matrix is diagonal, while keeping the original mean for each of them. I found this old article which seemed pretty ...
2
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1answer
66 views

Generating from $N_p(\mu,\Sigma)$ and Cholesky decomposition

I know how to generate random observation from $N_p(0,I)$ (applying Box-Muller transformation). But I was wondering how to simulate from $N_p(\mu,\Sigma)$(assuming $\Sigma$ to be pd). I started using ...
2
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2answers
246 views

Finding covariance given variance

Problem: Let $X$ and $Y$ be random variables such that $Var(X)=4$, $Cov(X,Y)=2.$ Find $Cov(3X,X+3Y)$. We know that: $$Cov(3X,X+3Y)=E(3X^2+9XY)-E(3X)E(X+3Y)$$ $$=3E(X^2)+9E(XY)-9(E(X))^2E(Y).$$ Also: $...
2
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2answers
55 views

Covariance of continuous functions, uniform and normal distribution

For X~Uniform(1, 9.9) and Y|X = x~Normal(1.4, x^2) What is Cov(X, Y) equal to? What I tried was: E[XY] - E[X]E[Y] Where E[X] = 5.45 and E[Y] = 1.4 But for E[XY] I'm a bit clueless. I've considered:...
2
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1answer
76 views

Example of “The eigenvalues of data covariance matrix, $\Phi^T\Phi$ measure the curvature of the likelihood function.”

I am reading PRML, Chapter 3.5.3, screen shot attached. I can understand the derivation and maths but hard to understand the meaning of "The eigenvalues of data co-variance, $\Phi^T\Phi$ matrix ...
2
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1answer
50 views

Showing $\operatorname{Cov}\left(\bar{X}_n,\frac{1}{n}\sum|X_i-\bar{X}_n|\right)=0$ for i.i.d standard normal $X_1,X_2,\ldots,X_n$

Show that for $X_1,X_2,\ldots,X_n$ i.i.d. standard normal, $$\operatorname{Cov}\left(\bar{X}_n,\frac{1}{n}\sum|X_i-\bar{X}_n|\right)=0$$ where $$\bar{X}_n = \frac{1}{n}\sum X_i$$ What I do first ...
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2answers
147 views

Variance of $X-Y$ cannot be negative

Suppose we have 2 non independent variable X and Y. Since $$Var(X-Y)= Var(X) + Var (Y) -2Cov(X,Y)$$ Would it be possible for the above to be negative in the case when $$Var(X) + Var(Y) < 2 Cov(X,...
2
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1answer
57 views

Is the series $X_j = Z_0 \cos(c j)$ stationary?

Let $Z$ be a gaussian white noise with mean $0$ and variance $1$ $c \in \mathbb{R}$ constant Is the time series stationary? I compute the mean and variance and they look constant, am I right? if so ...
2
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1answer
153 views

Expectancy of $X^TAX$ when $X$ is a matrix

To simplify, let's suppose that $E(X)=0$ and $A$ an idempotent and symmetrical matrix; that is $A^TA=A$. If $X$ is a vector, It is shown that $$E(X^TAX) = E\left(Tr(X^TAX)\right) = E\left(Tr(AXX^T)\...
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1answer
51 views

Compute the variance of $S = \sum\limits_{i = 1}^N X_i$, what did I do wrong?

This is a standard problem. Suppose we are given $$S = \sum\limits_{i = 1}^N X_i$$ where $X_i, i = 1, \ldots$ are iid random variables, with mean $m$, variance $\sigma^2$ and $N$ is an integer-...
2
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1answer
101 views

Show that $\operatorname{cov}(x,a + by) = b \operatorname{cov}(x,y)$

Let $x$ and $y$ be jointly distributed numeric variables and let $z=a+by$ , where $a$ and $b$ are constants. Show that $\operatorname{cov}(x,z)=b\, \operatorname{cov}(x,y)$. Here's what I have so far,...
2
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1answer
121 views

Computing eigenvalues of a specific block covariance matrix

In my studies of probability, I have recently come across the following problem on which i am stuck: Let x be an $ m $ dimensional random column vector, and let $ a $ be a random scalar variable, ...
2
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1answer
164 views

Is the matrix of exponential kernel with $L^1$ norm positive definite?

Given an exponential function $f(x) = \exp[-(q_1|x_1|+\dots+q_d|x_d|)]$ for $x\in \mathbf{R}^d$ with $q_i > 0 \;(i=1,\dots,d)$. For any distinct points $y_1,\dots,y_n\in\mathbf{R}^d$, define the $...
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1answer
114 views

Covariance and Variance

Given two random variables $X$ and $Y$ , I wish to find $Cov(X + Y, X − Y )$ assuming that $(a)$ $X$ and $Y$ are independent and $(b)$ $X$ and $Y$ are dependent & $Var(X) = Var(Y )$ I started ...
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1answer
220 views

Concentration inequality for covariance

Is there any concentration inequality for the covaraince of two scalar random variables? For example, how can I found a tight upper bound for the following probability? $$\Pr\left( {\left| {{\mathop{\...
2
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1answer
885 views

Covariance of the Kalman Filter innovation

I am trying to fully understand the derivation of the covariance of the innovation vector, however I am stucked conceptually at a point. I will show you my reasoning and where I am stuck (if someone ...
2
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1answer
94 views

Bounding covariance of random variables with point masses at zero using mixing coefficients

I have two random variables $X$ and $Y$ (defined on the same probability space) both with point masses at zero: $$\Pr(X = 0) > 0, \qquad \text{and} \qquad \Pr(Y = 0) > 0.$$ If it makes any ...
2
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1answer
162 views

Minimal Covariance of random variables.

Now I was wondering if you have some bernulli random variables $X_1, X_2, X_3,\dots X_n$. That distribute on some $1/2$ probability (Ber($1/2$)), and all of their Cov are equal, meaning that $\text{...
2
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1answer
204 views

Covariance stationarity of AR in time series

I have an extremely basic question of time series on covariance stationarity. Consider $$ Y_t=\gamma Y_{t-2}+\epsilon_t $$ where $\epsilon_t$ is such that $E(\epsilon_t)=0$ $\forall t$, $Var(\...
2
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1answer
275 views

Covariance matrix for least squares solution to $Ax = b$ when both $A$ and $b$ have uncertainties

Consider a linear system $Ax = b$ Assuming that $(A^\top A)^{-1}$ is invertible. The least square solution would be: $ x = (A^\top A)^{-1}A^\top b $ I'm looking to form an expression for the ...
2
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1answer
45 views

Covariance of random variables (returns)

How large are the correlations between each pair of the securities? I only know how to calculate their correlations if I knew their joint (discrete) distributions,since I need $E(K_jK_i), j \neq i$ . ...
2
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1answer
42 views

When should I search for the covariance matrix instead of the variance?

Suppose I have a random variable $X$ and $n$ realizations of this variable: $x_1, ..., x_n$. It seems clear to me in that case that if I am interested in knowing the variability I have in my data (...
2
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1answer
268 views

Why the covariance matrix of a matrix is the product $XX'$

Let a matrix of random observations such that $$\textbf{X}=\begin{bmatrix}x_{11}&x_{12}\\ x_{21}& x_{22}\end{bmatrix}$$ where $x_{jk}$ is the jth measure of the kth variable.Each column ...
2
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1answer
916 views

Intuition why Eigenvector of Covariance matrix points into direction of maximum variance

In context of principal component analysis, I am using the Eigenvectors of the Covariance matrix to project the data. I am able to prove that the Eigenvector of the Covariance Matrix is the direction ...
2
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1answer
620 views

Determine lambda parameter of exponential distribution from covariance

Good morning, Given an exponential distribution such that $X$ ~ $Exp(\lambda)$ with $Sn = \sum_{i=1}^n Xi$. Given that $Cov(S_{31}, S_{57}) = 31 + 57$, find the lambda parameter of the exponential ...
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1answer
34 views

Covariance Question [closed]

$X$ and $Y$ are two independent variables, with variances $\sigma^2_x$ and $\sigma^2_y$ respectively. Two other variables $W$ and $V$ are defined by $W=X+Y$ and $V=X-Y$. Find $Cov(X,V)$ and $Cov(W,...
2
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1answer
542 views

Two i.i.d Random Variables : Is covariance between a function (applied to one R.V) with the the R.V. the same for both Random Variables?

Let $X,Y$, both $$\Omega \to \mathbb{R}^k$$ be i.i.d random variables. Let $f$ be a function from $\mathbb{R}^k \to \mathbb{R}$. Is $\operatorname{cov}(f(X), X) = \operatorname{cov}(f(Y),Y)$? If we ...
2
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2answers
1k views

Find the covariance of $X$ and $Y$ given $Z = 3x - 2y$ and the standard deviation of $x, y$ and $z$.

This is the exact problem: Suppose that $X, Y$ are random variables with $Sx =2, Sy = 3$. Let $Z = 3X - 2Y$, and assume that $Sz = 6$. Find the covariance, $\text{cov}(X, Y)$. I have equations for ...
2
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1answer
22 views

How do I write the square of covariance?

Should it be $\mathrm{cov}^{2}(X,Y)$ or should it be $\mathrm{cov}(X,Y)^2$ or $(\mathrm{cov}(X,Y))^2$ or something completely different? Thank you.
2
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1answer
330 views

Question about creating $2\times 2$ covariance matrix with call option?

I'm completely stuck on how to do this problem. How can you go about calculating the variance of $Y$ and the covariance between $X$ and $Y$? I'm not sure how to use the information given to solve this ...
2
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1answer
77 views

Covariance and correlation, and how are they related?

I get that corellation is the covariance divided by the multiplie variance of the two, uh, things. What i don't get is why they are divided by the multiplied variance, and why that limits the value ...
2
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1answer
799 views

Variance of a sum of dependent random variables

Let's assume we have a random variable $z$ which really is a sum of two correlated variables: $$z \sim x + \operatorname{Bin}(x, p),$$ where $p$ is a constant. $x \sim \operatorname{Poiss}(\lambda)$....
2
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1answer
452 views

What is the covariance of two dependent normal distributed random variables

Problem is about getting the covariance of two random variables that are not independent: $\operatorname{cov}(\tilde{x}\mid(\tilde{y}=y),\tilde{x})= \text{ ?}$ $$\tilde{x}\sim N(\mu_1,\sigma^2_1)$$ $...
2
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3answers
553 views

How to calculate $V(X+Y)$ with $X$,$Y$ dependent?

I want to calculate the covariance of two dependent variables $X$ and $Y$ and I don't know the value of $V(X+Y)$, that is, the variance of $X+Y$. I know how the quantities relate to each other: $$V(...
2
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1answer
249 views

Law of large numbers with correlated variables

My sample is a series of measurements of the variable $x$. Measurement t, $x_{t}$, is correlated with $x_{t-n}$. However, as n tends to infinite the correlation tends to zero. If the sample grows, ...
2
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1answer
184 views

Expected Value and variance of a max randomized stocks

Hey guys I have been working on a probability and expected value/variance problem and the problem is: Each day the price of a stock in the market is a random number between 0 and 1 independently of ...
2
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1answer
84 views

Covariance and Independence problem

The Random Variables $X$ and $Y$ can each take on only two values. Show that if $Cov(X,Y)=0$, then $X$ and $Y$ are independent. One can see that the distributions take the form: $P(X=x_1)=p$ and $P(...
2
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1answer
144 views

Find marginal distribution of $Y$ where $Y\mid X$ is $N(a_1+a_2X,\sigma_1^2)$ and $X$ is $N(\mu,\sigma^2)$?

Let a random variable $X$ be normal $N(\mu,\sigma^2)$ and let the conditional distribution of $Y$ given $X$ be normal $N(a_1+a_2X,\sigma_1^2)$. a)Find the joint probability density function of $X$ ...
2
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1answer
104 views

Find the covariance of $Y_1$ and $Y_2$

I had a statistics question I was hoping for help on: Let $Y_1$ and $Y_2$ be discrete random variables with join probability function: $$f(x,y) = \begin{cases} \dfrac{y_1 + 2y_2}{18} & \text{if $...
2
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2answers
919 views

General sufficient condition for independence of these two random Variables.

I need to state and prove a general sufficient condition on(a,b,c) for independence of two random Variables. We have that $a,b$ and $c$ are real numbers and the random variables are below: $$ Y_1=...
2
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1answer
2k views

Covariance matrix of two matrices- how to calculate

In maximum covariance analysis, to extract correlated columns, it is asked to calculate the covariance matrix. For two vectors, corvariance matrix is understood, COV(v1,v2) = v1*v2' How do I ...
2
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1answer
1k views

Finding the ACVF

let $X_t = 0.5X_{t-1} + Z_t$ where $Z_t$ ~ $ WN(0,\sigma^2)$ I want to find the ACVF of both $X_t$ and $Z_t$, but I am a little bit confused. Say for $X_t$ $$\gamma(h) = COV(0.5X_{t-1} + Z_t, 0.5X_{t-...
2
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1answer
147 views

Evaluate this covariance matrix.

Edit: I have added an approach provided by @GiannisChantas. It would be great (and much appreciated) if someone could check if this approach is correct! I have also added a secondary question for ...
2
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1answer
525 views

Non-linear least squares with two dependent variables

I have data in the form $(t_i,x_i,y_i)$, i.e. position in 2D as a function of time. I have non-linear equations which I want to fit to the data. They give me a position $(X,Y)$ as a function of time ...
2
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2answers
59 views

probability, expectation, variance

A 10-digit long number is picked randomly and each digit's pick is independent and has an equal probability of being picked (1/9 because there's digits 1 to 9). Let $X = \#\{\text{missing digits}\}$ ...
2
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1answer
204 views

Calculating Covariance matrix of higher dimensions

I have the following matrix: $$A = \begin{bmatrix} 1 & 2\\ 3& 4 \end{bmatrix}$$ I therefore compute the Covariance matrix using the following: $$ \sum_{i=1}^{N} \frac{(x_{i} - u_{i})({y_{...
2
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1answer
47 views

Showing correlation is between $-1$ and $1$

If $X_1$ and $X_2$ are random variables, then $E(X_1^2)E(X_2^2) \geq [E(X_1X_2)]^2$ by Schwarz's inequality. Use this fact to show that $-1 \leq \rho_{12} \leq 1$. I know that $\rho_{12} = \frac{(E(...
2
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0answers
24 views

Covariance of errors $\text{cov}(\hat{e_{ij}},\hat{e_{i\ell}})$ in Two-Way Anova model

Exercise : Consider the Two-Way Anova model $Y_{ij} = \mu + a_i + b_j + e_{ij}$ with $i = 1, \dots, p$ and $j=1,\dots,q$. Show that : $$\text{cov}(\hat{e_{ij}},\hat{e_{i\ell}}) = -\sigma^2\left(\...
2
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0answers
32 views

Expectation of the product of three normal variables

Let $(X_1, X_2, X_3)\sim N(\mu,\Sigma)$ be a three-dimensional random variable where each coordinates are dependent (i.e. $\Sigma$ has non-zero values outside of its diagonal) I want to know how to ...
2
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0answers
46 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)=\...