# Tagged Questions

For questions about correlation of two random variables. Use it with [tag: random-variables] and [tag: probability].

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### How to calculate the correlation between two ratings

Suppose x and z are two persons who rate a classical piece of music y on a five point scale. x rates y $3$ out of $5$ z also rates y $3$ out of $5$ I need to know how much the ratings of x ...
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### Proof that correlation coefficient squared equals the coefficient of determination

Hi I as the title says I'm looking at the proof that $r^2$ = $R^2$ in the case of simple linear regression, but I don't understand one part. There are different versions of the proof, but in most of ...
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### OLS standard error that corrects for autocorrelation but not heteroskedasticity

Question: By mapping the OLS regression into the GMM framework, write the formula for the standard error of the OLS regression coefficients that corrects for autocorrelation but not heteroskedasticity....
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### Expectation and convolution question.

I am learning in an image processing course, and the professor did the following: As part of a derivation, has this: What I do not understand, is how he was able to remove $r(i,j)$ to the 'outside'....
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### correlation estimator

Suppose I have independent variables $X$ and $Y$ which follows exponential distribution with parameter $\lambda$. I want to find the variance of correlation estimator $\hat{\rho}$ which is defined as: ...
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### First and second moments of two correlated functions

I am trying to find the first and second moments of the following: $k=mk^{-b}$ where $m \sim U[a,b]$ (discrete) and $k^{-b}$ is a power law distribution I know how to find the first and second ...
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### Generate Correlated Normals

I want to generate normals $X,Y,Z$ with the correlation matrix $R$ but with means $0, 1, 2$ and variances $4, 16, 25$ respectively. How can I do this? Is it possible to apply Cholesky?
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### Reducing sequential correlations in Metropolis Algorithm

In our last lab, we use MCMC method to simulate a walker walking in the phase space. Using the Metropolis method, a walker at its currect position will sample another point inside a cube (centered at ...
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### What is the difference between Gaussian White noise and $iid$ noise and how can I check?

If I understand correctly, a series {$X_t$} is $iid$ noise if there is no trend or seasonal component and the observations {$x_t$} are independent and identically distributed with zero mean, while a ...
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### Why does Correlation Coefficient concern about the mean of the vector?

$$r = \frac {\sum_{i=1}^n (X_i-\bar X)(Y_i-\bar Y)}{\sqrt{\sum_{i=1}^n(Xi-\bar X)^2} \sqrt{\sum_{i=1}^n(Y_i-\bar Y)^2}}$$ This is exactly the $\cos$ of degree of the angle between vector $X-\bar X$...
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### How to find most correlated items?

[Complete noob here, apologies in advance] I have a table which contains a value for each other element in the table. I want to find out what the clusters are of related values (for instance, of ...
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### Are there any errors in my summary of stationarity? and some more questions.

I've posted questions about stationarity, but I cannot get answers satisfying me because of my vague question. Thus, I read more times about definitions about stationarity, summed them up, and brought ...
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### Pearson correlation of neural responses with it's linear estimation

I am trying to anderstand the following fact: Suppose I have a linear estimation of a stimulus: $\hat{s} = \mathbf{w}^T(\mathbf{r} - \mathbf{f}(s_0)) + s_0$ where $\mathbf{w}$ is a vector of ...
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### What do the eigenvalues of a correlation matrix represent?

I was wondering if there was any special meaning to the eigenvalues/eigenvectors of a correlation matrix. I get what they mean in a covariance matrix, and how that relates to PCA, though. Can you do ...
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### Meaning of horizontal bar in old formula (paranthesis?)

When reading an old paper from 1921* I find formulas like: $\rho + \frac{\rho(1- \rho^2)}{2\overline{n - 1}} \big( 1+ \frac{9 - 14\rho^2}{6\overline{n-1}} \big)$ which is said to be the median of ...
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### Proof that space of correlation matrices is compact

An $n\times n$ real symmetric matrix is a correlation matrix, if it is positive-semidefinite and all its diagonal entries equal 1. According to most references it is easy to see that the space of ...
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### Problems about Farlie-Morgenstern family of bivariate CDFs

Hi I am trying to solve the following problem: Let $F_X:\mathbb{R}\to[0,1]$ and $F_Y:\mathbb{R}\to[0,1]$ be unnivariate Cumulative Distribution Functions (CDFs) and suppose $-1\le\alpha\le 1$. Define ...
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### Measuring correlation of two time series

I have two sets of annual time series (employment growth from 2 different sources), which I display by using index. I would like to measure somehow whether series 1 (which are more timely) can be used ...
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### Show covariance matrix must have condition number larger than correlation matrix

The inverse of covariance matrix can be used to find the conditional independencies among variables. This inverse is more sensitive to changes then the correlation matrix. As a result two samples with ...
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### General correlation between function with itself and other input data

I've collected data for a function F = f(g(x)), for different function shapes g(x). The goal is to predict values of ...
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### Gaussian Copula Function

I have a question that I believe I know the answer to, but I would like to double check. Is it possible to use a copula function to calculate the joint probability of three or more variables? I'm ...
How can I show that $\dfrac{\hat{\rho } \sqrt{N-2}}{\sqrt{1-\hat{\rho}^2}}$ has a t-student distribution with $N-2$ degrees of freedom. I think I have to write it as a quotient of a normal $(0,1)$ ...