# Questions tagged [covariance]

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

1,086 questions
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### Determinants of the covariance between two random variables

I want to know if the covariance between two random variables is always determined by the two terms of its formula or if it can be only determined by the E(XY). If E(Y)=0, then Cov(X,Y)=E(XY)-E(X)E(Y)...
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### Given that $X_{i+1} = \rho X_{i}$, determine the dispersion matrix $Var[\textbf{X}]$

If $X_{1},X_{2},\ldots,X_{n}$ are random variables and $X_{i+1} = \rho X_{i}$ $(i = 1,2,\ldots,n)$, where $\rho$ is constant, and $var[X_{1}] = \sigma^{2}$, find $Var[\textbf{X}]$. MY ATTEMPT ...
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### Given two random vectors, determine the dispersion matrix $Var[\textbf{X}]$.

Let $\textbf{X} = (X_{1},X_{2},\ldots,X_{n})^{\prime}$ be a vector of random variables, and let $Y_{1} = X_{1}$ and $Y_{i} = X_{i}-X_{i-1}$ where $i = 2,3,\ldots,n$. If $Y_{i}$ are mutually ...
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### Calculate covariance given correlation, problem with percentages

The question is: find the covariance of ABC stock returns with the original portfolio returns. Pretty straightforward. However I get confused working between percentages and units. The ...
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### Covariance matrix of uniform distribution in $L^p$ Euclidean ball [on hold]

Let $$X=\operatorname{Unif}\left(\left\{x:\ \sum_{i=1}^n |x_i|^p\le 1\right\}\right)$$ Is there some method to calculate the covariance matrix of $X$? Thank you!
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### Using Cholesky decomposition to compute covariance matrix determinant

I am reading through this paper to try and code the model myself. The specifics of the paper don't matter, however in the authors matlab code I noticed they use a Cholesky decomposition instead of ...
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### truncation degree of decomposed covariance matrix

I have a covariance matrix of a standardized data set. Doing a singular value decomposition i find near zero singular values and would therefore like to truncate it. I know of Picard plots which ...
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### Covariance of square root for two bins of a multinomial

Take $(X_1, \dots, X_k) \sim Multinomial(n, (p_1, \dots, p_k))$. Do we have a closed form expression for $\mathbb{E}[\sqrt{X_i X_j}], i\neq j$ ?
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### A box contains 3 red balls and 2 white balls

A box contains 3 red balls and 2 white balls. Two balls are picked randomly from the box without replacement. The random variable 𝑿 is the number of red balls, and 𝒀 is the number of white balls. ...
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### Normal Random Variables that all correlate with a single time series but not necessarily with each other

I have a sequence of normally distributed random variables. Let's call it $S_1$. I want to generate 4 more series, each of which has its own correlation with $S_1$ and its own variance. Let's call ...
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### Scalar product induced by covariance matrix

Suppose that $n$-dimensional random vector $Y$ has covariance matrix $\Sigma$. It is well known that for any $a\in\mathbb{R}^n$ we have \begin{align} var(a^TY)=a^T\Sigma a. \end{align} Is there any ...
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### Finding $\text{Cov}(X,Y)$ when $(X,Y)$ has joint density $\frac{1}{2}\sin(x+y)\mathbf1_{0\le x,y\le\pi/2}$

Joint probability density: \begin{equation} P_{x,y}(x,y) \begin{cases} \frac{1}{2}\sin(x + y) & , \text{if}\ 0\leq x \leq \frac{\pi}{2}, 0 \leq y \leq \frac{\pi}{2} \\ 0 & ...
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### Variance and covariances from linear mixed model for power simulation using R

I am working with longitudinal data where the outcome is the number of steps per minute. My LMM fit would look like: ...
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### Constructing a probability measure on the Hypercube with given moments

Let $H = [-1, 1]^d$ be the $d$-dimensional hypercube, and let $\mu \in \text{int} H$. Under these conditions, I can explicitly construct a tractable probability measure $P$, supported on on $H$, ...
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### How to show that the error variance of the best linear predictor is inferior to the proportional predictor?

Let's consider the 1D case. How do we prove that the error variance of the Best Linear Predictor (BLP) is inferior than the Proportional Predictor (i.e. the Linear Predictor without the intercept)? ...
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### Stretching the covariance of a daily change to the year

I'm diving into financial mathematics and have calculated a matrix that gives me the daily change of four given securities: The start of it looks like this: My Tutorial is now calculating the ...
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### We have an urn with 6 red balls and 4 green balls.

We have an urn with 6 red balls and 4 green balls. We draw balls from the urn one by one without replacement, noting the order of the colors, until the urn is empty. Let X be the number of red balls ...
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### Testing cross-covariance in the residuals of a VAR(p) model

Suppose I have a vector autoregressive model of order $p$: $$y_t = c + A_1 y_{t-1} + ... + A_p y_{t-p} + u_t$$, where $y_t$ is a $K\times 1$ vector and $A_i$ are $K\times K$ matrices. We assume the ...
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### Prove a time series to be NOT identically independent distributed

I am trying to prove that this time series (given that $X_{t}$ and $M_{t}$ are iid and independent of each other) $$Y_{t} = X_{t}(1-X_{t-1})M_{t}$$ is not i.i.d, so my understanding is that I need ...