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Questions tagged [covariance]

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

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Is the positive part of a covariance stationary process also stationary?

I am wondering if it is possible to derive a result on the stationarity of the positive or negative part of a covariance stationary process. Namely, consider $\{ X_t \}, t=1,2,3,...,$ a covariance ...
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671 views

Covariance matrix and Gaussian i.i.d. random variables

I have a set $X = \left \{ X_i | i \in (1,n) \wedge X_i \text{ is a random variable} \right \} $ Does $\forall i \in (1,n ), X_i \text{ follows a normal distribution} $ implies that $\...
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48 views

Under what condition are $U$ and $V$ uncorrelated?

Let $X$ and $Y$ be independent random variables with finite variances, and let $U = X + Y$ and $V = XY$. Under what condition are $U$ and $V$ uncorrelated? MY ATTEMPT We say that two variables are ...
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Proof that $Cov(aX+b, Y+Z)=aCov(X,Y)+aCov(X,Z)$

I'd like to show that $Cov(aX+b, Y+Z)=aCov(X,Y)+aCov(X,Z)$. Therefore I use: $Cov(X,Y)=E(XY)-E(X)\cdot E(Y)$. So $Cov(aX+b, Y+Z)=$ $=E[(aX+b)(Y+Z)]-E(aX+b)E(Y+Z)$ $=E(aXY+aZ+bEY+bEZ)-[(aEX+b)(EY+...
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183 views

Does $\operatorname{Cov}(X,Y) = 0$ mean $\operatorname{Cov}(X,\log Y) = 0$?

Suppose $X,Y$ are positive random variables with $\operatorname{Cov}(X,Y)=0$. Define $Z= \log Y$. Does it necessarily follow that $\operatorname{Cov}(X,Z) = 0$? I know it's true for linear ...
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123 views

Variance and covariance inequality

Given a real-valued random variable $X$, is $$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$ true? Any pointers for how to tackle this problem would be immensely helpful.
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132 views

Why does the identity $\mathbb{E}(X) = \mathbb{E}\left(\int \mathbb{1}_{u \leq X}du\right)$ hold?

I'm reading on Hoeffding's covariance identity, the proof of which is neatly covered here, or, in a similar manner, in this MSE post, but I can't seem to fully understand the trick/property used there....
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13k views

Variance of X - Y

If X and Y are random variables with correlation coefficient 0.7, each of which has variance 6, what is the variance of X−Y? Enter your answer as a decimal. Using the information given, I was able ...
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3answers
58 views

How could I find the covariance for $X$ and $Y$ in this case?

If $X \sim U(-1, 1)$ (so $X$ is uniformly distributed between $-1$ and $1$) and $Y = X^2$, what is the covariance between $X$ and $Y$? Are they independent? So the formula for covariance is: $\...
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Compute $Cov(X+Y, \frac{X} {X+Y})$

Here's my problem: Assume that positive random variables $X$ and $Y$ are identically distributed with $E[X] = E[Y] = μ < ∞$, and not necessarily independent. Compute $$Cov\left(X+Y, \frac{X}{X+Y}\...
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Probability mean,variance and standard deviation formula confusion.

I have a confusion in the formula attached. Why and how are the two formulas equivalent ? sigma in the image is the standard deviation of a distribution...
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2answers
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Covariance matrix of an AR(1) model?

The covariance matrix of the values of the AR(1) model $X_t = \phi X_{t-1} + Z_t$ at times $t=1$ and $t=3$ is useful to find the best linear predictor of $X_2$ given $X_1$ and $X_3$. Let $W = (X_1, ...
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Covariance for stochastic variables

if $X$ and $Y$ are stochastic variables with $\operatorname{Var}(X)=1.34$ and $\operatorname{Cov}(X,Y) = 0.64$, find $\operatorname{Cov}(2X, 3X+2Y)$. No ideas on this one, as I don't see any way of ...
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1answer
356 views

Ito isometry with two independent Brownian motions

Let $V_t, W_t : [0,T] \times \Omega \to \mathbb R$ be indpendent Brownian motions defined up to time $T$ and let $X_t,Y_t : [0,T] \times \Omega \to \mathbb R$ be stochastic processes adapted with ...
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Why is finding $M$ eigenvectors on smaller matrix valid?

I am following this article on face recognition. In "calculating eigenfaces" section, the authors present a solution for the problem of calculating a very big matrix: Let $A_{N^2\times M}$ be an $M$ ...
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1answer
232 views

What is the covariance of $\min(X,Y)$ and $\min(X,Z)$ when $X$, $Y$, $Z$ are i.i.d. standard normal?

Ecological growth can be thought of as being determined by the most limiting factor, and I'd like to have a solution for the covariance of growth records subject to common and independent limitations. ...
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382 views

How to calculate variance of W? Find the probability distribution of W?

$W=Y-X$ I have figured out that $E(W)=0.3$ by using this formula $E(X+Y)=E(X)+E(Y)$. I tried using the same formula with $E(X^2)$ and $E(Y^2)$ to find $E(W^2)$. I also tried using $V(X+Y)=V(X)+V(...
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Variance of max - min of 2 exponential random variables

Suppose we have 2 random variables, $S\sim Exp(\lambda)$ and $T\sim Exp(\mu)$. Let $U=\min(S,T)$ and $V=\max(S,T)$. What is the variance of $W=V-U$? I calculated: $Var(U) = \frac{1}{(\lambda+\mu)^...
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1answer
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Finding $E(XY)$ given $E(X)$, $E(Y)$, $E(Y^2)$, $E(X|Y)$

So here is what I know: \begin{align} E(X) &= 13\\ E(Y) &= 2\\ E(X|Y) &= 3y + 7\\ E(Y^2) &= 8 \end{align} How do I find $E(XY)$? Thanks!
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Prove that $\operatorname{Cov}[X,E(Y|X)]=\operatorname{Cov}[X,Y]$

How can I prove that $\newcommand\cov{\operatorname{Cov}}\cov[X,E(Y|X)]=\cov[X,Y]$? I tried $\cov[X,E(Y|X)] = E[XE(Y|X)]-E(X)E[E(Y|X)] = E[XE(Y|X)]-E(X)E(Y)$ then I am stuck. How can $E[XE(Y|X)] = E(...
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1answer
2k views

Is cov(x,y) and cov(y,x) the same thing?

I am learning the concepts of covariance and covariance matrix. It seems to me that: Cov(x, y) = E((x - E(x))(y-E(y))) = E((y-E(y))(x-E(x))) = Cov(y,x) Is that the case? If so, why do we need to ...
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1answer
167 views

Computation of the covariance of a stochastic integral.

Set $X_t= \int_{0}^{t}\sqrt{s}\sin(B_s)dB_s, t\geq 0$. How can I compute de covariance between $X_t$ and $X_u$, $0\leq u \leq t$? I started using Itô isometry but I can´t go any further. Thanks.
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4answers
806 views

Why does a negative covariance reduce the variance of the sum of two dependent variables?

If I am interested in Var(X + Y)=Var(X) + Var(Y) + 2Cov(X,Y) where X and Y are dependent iid random variables, there is the possibility that the covariance could be negative, which would yield a ...
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2answers
239 views

How to find the Expectation and Covariance of a Process?

How would one find the expectation and thus the covariance of a process, say, $$X(t)=\alpha +B(t)-tB(1)$$ is the process and $t\in[0,1]$ and $\alpha\in\mathbb{R}$? When considering this process ...
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Can I bound the correlation of two random variables using the mutual information?

As correlation $\rho_{X,Y} := \frac{Cov(X,Y)}{\sigma_X \sigma_Y}$ sort of measures the linear dependence of two random variables, and mutual information $I(X; Y) := H(X) - H(X|Y)$ measures the ...
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499 views

What is the relationship between ellipse and hyperbola?

I am working with lines both in polar ($\rho-\theta$) and Cartesian co-ordinates. Initially, the line is given by $(\rho_0, \theta_0)$ and the covariance in the parameter space is $$R=\left(\begin{...
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1answer
347 views

Is $AB$ a covariance matrix?

Suppose we have two covariance matrices $A$ and $B$. They satisfy the condition $AB=BA$. Is $AB$ a covariance matrix? My answers: We can easily check that $(AB)'=B'A'=BA$, then $AB$ is symmetric. ...
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Covariance of multinomial distribution

Let $X = (X_1,\ldots, X_k)$ be multinomially distributed based upon $n$ trials with parameters $p_1,\ldots,p_k$ such that the sum of the parameters is equal to $1$. I am trying to find, for $i \neq j$,...
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Proof of Hoeffding's Covariance Identity

Let $X,Y$ be random variables such that $\text{Cov}\left(X,Y\right)$ is well defined, let $F\left(x,y\right)$ be the joint-CDF of $X,Y$ and let $F_{X}\left(x\right),F_{Y}\left(y\right)$ be ...
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1answer
58 views

Check if $Cov(X_1+X_2,X_1-X_2)=0$, i.e. if independent?

"Let $X_1$ and $X_2$ be independent, $N(0,1)$-distributed random variables. Show that $X_1+X_2$ and $X_1-X_2$ are independent." I know that for multivariate normal distributions independence can be ...
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1answer
584 views

Covariance between squared and exponential of Gaussian random variables

Assuming the random vector $[X \ \ Y]'$ follows a bivariate Gaussian distribution with mean $[\mu_X \ \ \mu_Y]'$, and covariance matrix $ \left[ \begin{array}{cc} \sigma_X ^2 & \sigma_Y \ \sigma_X ...
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1answer
110 views

Gaussian kernels for arbitrary metric spaces

Let $(I,d)$ be an arbitrary (pseudo-)metric space. Define the function $$c(i,i') := \exp\big( - d(i,i')^2 / 2 \big)$$ Is $c$ necessarily nonnegative-definite, hence a kernel function?
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942 views

mean and variance normalization of vectors

I have vectors $x \in \mathbb{R}^n$ and I expect some multivariate normal distribution. I want to normalize the vectors in such a way that $y = M (x - b)$ has mean zero ($\operatorname{E}[Y] = 0$) ...
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1answer
23 views

Find covariance of a random process

Let $Y_t = e^{-\alpha t}W_{\beta \ \exp({2\alpha t})}$, where $W_{s} \ \text {is Wiener prosses,} \quad 0\le t, \quad \alpha , \beta\in \mathbb R^1$. Find $\text{Cov}(Y_t , Y_s).$ Here is my ...
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1answer
16 views

Show that the autocovariance function depends on $s$ and $t$ only through their difference $\left|s-t\right|$

Consider the time series $$ x_t = \beta_1 + \beta_2t + w_t, $$ where $\beta_1, \beta_2$ are known constants and $w_t$ is a white noise process with variance $\sigma^2_w$. I want to show that the ...
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1answer
54 views

Finding the probability of two random variables being equal to 1

Question: A die is thrown $n+2$ times. After each throw a '$+$' is recorded for $4$, $5$, or $6$ and '$-$' for $1$,$2$, or $3$, the signs forming an ordered sequence. To each, except the first and ...
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2answers
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finding the common variance of a die throw

would appreciate your help with this question: a regular die is being thrown 21 times. we define: $x_1$ - the number of throws we obtained 1 or 2. $x_2$ - the number of throws we obtained 3,4,5,6. ...
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2answers
40 views

calculate $\operatorname{cov}(X,Y)$ from $f_{x,y}(x,y)$

I have the following density function: $$f_{x, y}(x, y) = \begin{cases}2 & 0\leq x\leq y \leq 1\\ 0 & \text{otherwise}\end{cases}$$ We know that $\operatorname{cov}(X,Y) = E[(Y - EY)(X - EX)]$...
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2answers
879 views

Central limit theorem for dependent random variables with covariance condition

Consider a sequence of identically distributed real-valued random variables $(X_i)_{i\in\mathbb{N}}$, with $\mathbb{E}\left[X_i\right]=0$ and $\mathbb{E}\left[X_i^2\right]=1$. Suppose that there ...
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1answer
85 views

Correlation function for ARMA($2$,$1$)

I have to derive analytic expressions for the autocorrelation function of this ARMA($2$,$1$) process: $$y_{t} = \varphi_{1}y_{t-1} + \varphi_{2}y_{t-2} + \varepsilon_{t} + \theta\varepsilon_{t-1}$$ ...
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2answers
30 views

Normal distribution, uncorrelated random variables

$Y$ is a random variable such that $$Y = \cases{X,\hspace{2mm} \mathrm{for}\hspace{1mm} |X| \leq c \\ \\-X,\hspace{2mm}\mathrm{for}\hspace{1mm}|X|>c,} $$ where $c \geq 0,$ and $X$ has the standard ...
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1answer
63 views

Covariance between centered and scaled normal entries of a random vector

From this post and following a tip by @Ian in the comments. If $X_1,\dots,X_n \sim \text{ i.i.d. } N(\mu,\sigma^2)$ with $\displaystyle \bar X= \frac{\sum_{i=1}^n X_i}{n},$ the covariance of the ...
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1answer
468 views

Variance of Chi Square Distribution as the Sum of Unit Normal Random Variables

Okay, so I am interested if there is a way to derive the variance for a Chi-Square distribution using the property that it is the sum of independent unit normal distributions squared. For example, ...
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1answer
47 views

Proof that sum over autocorrelations is -1/2

I am trying to understand a proof that shows, that the sum over the autocorrelation starting with lag=1 is always equal -1/2, for a stationay time series. The sum looks like this: $$ S_{\rm{afc}}=\...
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1answer
65 views

If Cov(A,B) = 0, what can be said about Cov(|A|, |B|)?

If I have two random variables $A$ and $B$ taking values in $[-1,1]$ (where both $1$ and $-1$ have some non-zero weight), and I know that $Cov(A,B) = 0$, can anything at all be said about $Cov(|A|, |B|...
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1answer
61 views

Proof $\rho(X,Y)=\pm1$, only for special cases

I'm having some difficulties with the following proof: Only onder special circumstances it can be the case that $\rho(X,Y)^1=\pm1$, and these circumstances are explored by considering the proof of ...
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1answer
740 views

What does it mean to minimize a matrix

In Gilbert Strang's 'Introduction to Applied Mathematics', in chapter 2.5 (Least Squares Estimation and the Kalman Filter), in proof 2I, he talks about 'minimizing the covariance matrix'. I can't ...
2
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1answer
106 views

covariance of random variables

Suppose X, Y, W are independent random variables such that X ∼ GAM(2,3), Y ∼ N(1,4) and W ∼ BIN(10,1/4). Let U = 2X − 3Y and V = Y − W . Find cov(U, V ). I know that cov(U, V) = E(U, V) - E(U)E(V). I'...
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1answer
2k views

Projection theorem for conditional probability

$\def\cov{\mathop{\mathrm{cov}}}\def\var{\mathop{\mathrm{var}}}$My professor uses something that he calls the "projection theorem", to get rid of the condition in conditional probabilities (...
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1answer
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 ...