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

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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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2answers
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Is this altered formula for correlation still bounded by $-1$ and $1$?

Recall that $$ ‐1 \le \text{corr}(X,Y) = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \le 1 $$ The proof for this bound uses the Cauchy Schwarz inequality, and I've been trying to wrap my head around ...
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3answers
33 views

Is it true that $\frac{E[(|X - E[X]|)(|Y - E[Y]|)]}{\sigma_X \sigma_Y} = 1$?

Consider the well-known fact that correlation is bounded between $-1$ and $1$: $$ -1 \le \text{corr}(X,Y) = \frac{E[(X - E[X])(Y - E[Y])]}{\sigma_X \sigma_Y} \le 1. $$ I've been trying to wrap my ...
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20 views

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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26 views

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 ...
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1answer
29 views

Correlation and Linear Regression

I'm tasked with this question but unable to proceed on. Q: Calculate the linear product moment correlation coefficient between x and m for these samples: $$ \Sigma x=205,\\ \Sigma m=1240, \\ \Sigma ...
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16 views

Coupling Brownian Motions

I want to simulate three freight rate indices which are naturally correlated. The freight rate dynamics ($X$) can be modeled as a geometric Brownian motion: $dX_{t} = \mu X_{t}dt + \sigma ...
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1answer
21 views

(a) Calculate the correlation between X and Y, Corr(X,Y).

Suppose X and Y are random variables, such that $$E(X)=5, E(Y)=3, E(X^2)=26, E(Y^2)=13, E(XY)=10$$ We used the equation enter image description here and got $$r=\frac{10n ...
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0answers
20 views

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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1answer
17 views

Autocorrelation of heaviside functions

I'm trying to find the expression that describes the auto-correlation $r_{xx}(\tau)$ of two heaviside functions $u(t)$. I was told that the result must be $1/2$, which makes total sense, as the power ...
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1answer
34 views

Spearmans Rank, why does it work?

Looking at spearmans rank, can someone explain how the forumula works, is their anything intuative about it?
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17 views

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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0answers
42 views

centering two variables $X$ and $Z$ makes $cov (X,XZ) = 0$

I've read that centering two normal (or symmetrical) variables $X$ and $Z$ affects correlation of centered $X$ with interaction term $X\cdot Z$ in such way, that this correlation $cor(X-EX, X\cdot Z)$ ...
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0answers
22 views

Calculation of a Autocorrelation function and Power spectral density

A sample of a random process is given as: $$ x(t) = Acos(2\pi f_0t) + Bw(t) $$ where w(t) is a white noise process with 0 mean and a power spectral density of N0/2, and f, A and B are constants. ...
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1answer
708 views

Pearson correlation and metric properties

Assuming that the data set was $z$-standardized to zero mean and unit variance (also assuming that it does not contain constant vectors). Then Pearson's r reduces to Covariance: $$\rho(X,Y) := ...
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1answer
16 views

Which variables to use in regression

If I have variables x1,x2,x3,and x4 that have correlation coefficients $-0.9, -0.5, 0.5,$ and $0.9$ to another variable y, what is the effect of choosing different combinations of them in a ...
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1answer
1k views

Autocorrelation and spectral density in MATLAB

This question is twofold. We have an LTI system that is a first degree Butterworth LP filter with the power TF where fu = 110Hz and ...
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0answers
11 views

Cross correlation of (gaussian distributed) singals with the mean signal gives log-normal density function

The following is my question: I have signals that contains noise, they are of the following form see Figure 1. Then I take the mean signal of all these signals (identical in length and shape). Just ...
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0answers
9 views

Evaluating the spectral density of generated noise through the autocovariance

Arguably more of a question for the signal processing page, but I feel it could also belong here. I'm working on generating noise signals $X(t)$ (with $t \in \left[0,T\right]$ with step size $\delta ...
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1answer
15 views

Variance, Covariance, and Correlation answer check

Two random variables, $Y$ and $Z$: $Y = 0.5+0.6X$ $Z = 0.2+0.3X$ where $X$ is another random variable. You can treat the variance $var(X)$ as a given constant. It may help to give $var(X)$ a name, ...
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0answers
16 views

Covariance matrix of random vector of vectors

I am a beginner in statistics and tried to research my question online without much success. Motivation: I am working on an undergraduate project in cosmology. My problem involves several ...
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1answer
740 views

Probability of three events occurring given correlation?

I am facing a problem that I cannot find the answer to. I have three variables, A, B and C. There are only two possibilities for each of these, A either happens or it does not, B happens or it does ...
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2answers
28 views

How can I determine the best relationship for 3 variables, given several data points?

What is the best way to determine the relationship for three apparently related variables? The relationship does not appear to be linear, and may follow a combination of non-linear functions. I have ...
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1answer
57 views

Multiple regression and hypothesis test $H_0$:$\beta_2=0$

Multiple regression model $H_0$:$\beta_2=0$, $H_1$:$\beta_2 \neq 0$ where $\beta_2$ is the vector of elements ($\beta_2, \beta_3, \dots, \beta_k$) and $\beta$ is slope of regression line. Why it is ...
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1answer
46 views

Correlation in a series of 1s and 0s [duplicate]

I have a series (in reality I have several series) of 1s and 0s (success and failures) and an "estimated" success-probability as well as the actual success-rate (I can count the 1s in my set). I want ...
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5answers
66 views

Suppose $X, Y$ are random variables with the equal variance. Show that $X-Y$ and $X+Y$ are uncorrelated.

Suppose that $X$ and $Y$ are random variables with the equal variance. Show that $X-Y$ and $X+Y$ are uncorrelated. I get I should use the equation $$E[XY] = E[X]E[Y]$$ For the first part I get ...
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0answers
29 views

Correlation in Bernoulli trial

I have a large dataset of tennis-points (who served and if they won etc). Now I would like to check if the points have any correlation with each other (does winning/losing the previous point affect ...
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1answer
25 views

cov(X,XY)? if X,Y is not independent

For two normal random variables , $X$ and $Y$ whose mean are not zero, If $ cov(X,Y) $ is given as $\sigma_{XY}^2 $ , are there any simple way to calculate $ cov(X,XY) ?$
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1answer
32 views

Interpreting the scatter plots of two random variables

Suppose I sample random variables A and B from normal distributions. When I do a scatter plot of these two variables I see a ...
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2answers
31 views

Correlation of sum of independent variables with its parts. if Z=X+Y, what is Cor(Z,X)?

If $Z = X + Y$, where $X$ & $Y$ are independent random variables, is there some formula to work out $\rho(Z,X)$, based on $\sigma_X$, $\sigma_Y$? For example, I've noticed that for $\sigma_X$ = ...
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1answer
30 views

finding correlation coefficient given conditional expectations

Given $Y1$ and $Y2$ have a bivariate normal distribution where $E(Y1|Y2)=4.7-0.16Y2$ and $E(Y2|Y1)=0.8-Y1$ and conditional variance is 3.64. How can I find the correlation. I have tried the ...
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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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Taylor expansion of Pearson's sample correlation coefficient

I'm trying to find Taylor series expansion of Pearson's sample correlation coefficient and I don't really know how to do it. Any ideas?
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1answer
19 views

Assess two matrixes overlapping

I have two binary matrixes, of the same size (e.j. 5000x5000). Those matrixes represent the same area, divided in cells of the same size. Each cell of one matrix can be true or false, meaning some ...
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Correlation coefficient function (function of time) has obvious hape but weak absolute value

I am doing a research about a 2D space. I am looking at the angle alinement between the eigenvalues of stress and strain respect to time. When I plot the angle alinement vs. time, it showed an ...
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1answer
36 views

Ratio laying within the confidence interval still being depicted as having an influence?

I keep seeing this in research papers. The researchers claim that there is a positive correlation between A and B then subsequently show that they odds ratio/sample mean etc. is IN the confidence ...
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1answer
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Correlation coeffcient expressed in a different form

If $X_j = a + bX_i$ and $X_i = a' + b'X_j$ how does one show that the correlation coefficient can be written as: $\rho_{i,j} = b \frac{\sigma_i}{\sigma_j}=b'\frac{\sigma_j}{\sigma_i}$ ?
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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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1answer
25 views

Distinction between correlation coefficient and coefficient of determination

In my stats class, I am learning about correlation coefficient and coefficient of determination. I dont understand what the difference is between them. there are $r,\,$ $r^2$ and $R^2$. $r^2$ and ...
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1answer
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Correlation between a random variable and its rank

Let $X_1,\ldots,X_n$ be a random sample from $U(0,1)$ and $X_{(1)}<\ldots<X_{(n)}$ be the corresponding order statistics. Define, $$ R(X_1) = r\quad \text{if}\quad X_{(r)} = X_1;\quad r = ...
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2answers
38 views

Generate two negatively correlated data in excel

Let's say that we have two prices that are negatively correlated to each other, for instance we have price $p_1$ and we want to generate negatively correlated price $p_2$ with the following ...
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0answers
27 views

Determining whether or not random variables are correlated

I'm working on the following problem: Consider random variables $X$ and $Y$ such that exactly one of them is equal to $0$. The other then takes the value $1$ or $-1$ with equal probability ...
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2answers
62 views

question about correlation of variables

Here is an interview question I had and cannot figure out how to solve it. Any hint? Let $X$, $Y$, $Z$ be 3 random variables such that $\mathsf{Corr}(X, Y)=0.9$ and $\mathsf{Corr}(Y, Z)=0.8$. What ...
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0answers
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zipf and lognormal with a particular correlation

I have been struggling on how to generate a correlated zipf and lognormal distribution. I want to generate a set of data ,say,$(X,Y)$,where $X$ is the popularity of file described by ...
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1answer
37 views

Necessary sufficient condition for correlation between last 2 out of 3 normal random variables.

Consider $X,Y,Z$ all standard normal random variables. Now I also want to have some correlation between them. Lets denote these $\rho_{XY},\rho_{YZ}$ and $\rho_{XZ}$. Surely I cannot choose them ...
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0answers
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Calculating the coefficient of concordance for vague data

I'm trying to adapt a formula that calculates the coefficient of concordance for vague data. The paper that describes it is here ...
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310 views

How to find a correlation and whether it is statistically significant

I have $2$ columns with numers in them, $x$ and $y$. I'd like to find out what $y$ could be if $x = n$ and how strong the correlation is, if it's statistically significant or not. How do I do that? ...
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3answers
2k views

Is correlation (in some sense) transitive?

If we know that A has some correlation with B ($\rho_{AB}$), and that B has some with C ($\rho_{BC}$), is there something we know to say about the correlation between A and C ($\rho_{AC}$)? Thanks.
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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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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 ...