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

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25 views

Weighted Average of Correlation Matrix

Let $R$ and $Q$ be two correlation matrices of the same size and let $p\in[0,1]$. I'm trying to show that $pR+(1-p)Q$ is still a correlation matrix. I claim that $\sqrt pX+\sqrt{1-p}Y$ is a vector ...
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1answer
33 views

Joint PDF Correlation

In the problem I am given $f(x,y)=2,\ 0 < x < y,\ 0 < y <1$. I'm trying to find the correlation $\rho$ which I know is equal to $$\rho = \frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}$$ ...
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1answer
63 views

Computing the expected value of the product of two discrete variables

I didn't know why I compute $E(XY)$ wrongly. $$X=(1, 2, 0.5, -1),\qquad Y=(-2, 1, -0.5, 2).$$ $$E(XY) = \frac{-2 + 2 -0.25 -2}{4} = -0.5625\text{ (incorrect)}$$ because $\operatorname{Cov}(X,Y)=-0....
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1answer
44 views

Size of sample and correlation coefficient

$X$ and $Y$'s correlation coefficient is $r=0.5$. What is the size of sample when the correlation is significant at $\alpha=0.05$ with two sided test? Is there a more "formal" way to solve this ...
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239 views

Partial proof for correlation coefficient formula?

I've been working to prove the formula for the correlation coefficient, since asking my last question yesterday (Meaning of denominator in correlation?). If this post in any way violates any ...
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2answers
455 views

Meaning of denominator in correlation?

I can't quite grasp the meaning of the denominator in the correlation coefficient. $$\frac{\sum(X - \bar X)(Y-\bar Y)}{\sqrt {\sum (X-\bar X)^2\sum(Y-\bar Y)^2}}$$ What exactly am I dividing with, ...
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1answer
104 views

Correlation coefficient and Expectation of two dimensional normal distribution.

Random variable (X,Y) is normally distributed. Conditional expectations are $E(X|Y=y)=0.25y + 2$ $E(Y|X=x)=x-2$ How can i determine correlation coefficient and when that is known, the expectations ...
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1answer
88 views

rewriting formula containing covariance and variances

just trying to follow a formula. the equation starts of as follows, 1 = sum( xi * (cov(ri, r) / sigma^2(r) ) please note i's are subscripts then next line ...
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1answer
67 views

What is the expectation of the product of dependent, normal random variables

Question: Let's say I have $X \sim N(\mu_1, \sigma) $ and $Y \sim N(\mu_2, \sigma) $. I know that $ cor(X,Y) = \rho $. What is $E(XY)$? What I've tried Based on a similar question where X and Y are ...
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187 views

relationship between multiplication and correlation

is there a deep interpretation of multiplication as correlation? is this in some sense the most fundamental way to "combine" objects (eg numbers) into products? my reasons for asking are that the ...
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1answer
52 views

Correlation between 3d images and their slices

I work in the field of the image processing and I need to compare results of my algorithm with a gold standart results. For this purpose I calculate the Pearson correlation coefficient between the ...
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2answers
46 views

Show $E\left(\mathbf{X}_i \otimes \mathbf{u}_i\right)=\mathbf{0}$ implies $E\left(\mathbf{X}_i^{\top}\mathbf{G}\mathbf{u}_i\right)=\mathbf{0}$

Let $\mathbf{X}_i$ be a $G \times K$ random matrix, and let $\mathbf{u}_i$ be a $G \times 1$ random vector, and suppose we have a sample of $i=1,\ldots,N$ of each. Suppose the following condition ...
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0answers
143 views

Definition of Autocorrelation Function (ACF)

For a weakly stationary time series $\left\{r_{t}\right\}$, the definition of ACF is (from Ruey Tsay's "Analysis of Financial Time Series") $$ \rho_l=\frac{Cov(r_t,r_{t-l})}{\sqrt{Var(r_t)Var(r_{t-l})}...
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204 views

Finding a relative error measure on a data set proportional to another

I have a set of exact data points $\mathcal{X}=\{X_i\}$ and another approximate one $\mathcal{Y}=\{Y_i\}$ where there is a correspondence between $X_i$ and $Y_i$ for all $i$. If $\mathcal{Y}$ was ...
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13 views

Proving $Corr(\hat{e}_{ij}, \hat{e}_{jk}) = \frac{-1}{n_i-1}$ for $ j \neq k$

For the model of a single factor experiment: $y_{ij}= \mu + \alpha_i + e_{ij}$, $(1 \leq i \leq a, 1 \leq j \leq n_i)$, where a = the number of treatments, $n_i$ = the number of experimental units ...
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2answers
177 views

The inverse of AR structure correlation matrix / Kac-Murdock-Szeg ̈o matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots &\...
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1answer
558 views

What does the multiplication of standard deviation of two variables gives?

If we need to find the correlation between two variables it is given by the formula - co variance of two variables divided by the multiplication of Standard deviation of the two variables. My ...
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1answer
40 views

Question on the correlation between two dependant variables

I'm working on this question and it's stumping me. Let Sn = X1 + ... + Xn (with n>=1) be a random walk with X1,...,Xn be iid RV's. E(Xk)=mu Var(Xk)=sigma^2. Find the covariance of Sn and Sm Can ...
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1answer
74 views

Combination problem: random selection in a group

A scientific committee of 4 persons is to be randomly selected from a group consisting of 3 biologists, 3 physicists and 4 mathematicians. Let X denote the number of biologists, Y the number of ...
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37 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = \...
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0answers
48 views

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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1answer
35 views

Is $d(i,j) = 1-\textrm{corr}(i,j)$ a metric?

I need to make sure that this function is a metric: $d(i,j) = 1-\textrm{corr}(i,j)$ where $\textrm{corr}(x,y)$ is the Pearson correlation coefficient which ranges from $[-1,1]$. With this scaling I ...
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40 views

Bounding the Correlation Coefficient

Let us assume we have two random variables $X$ and $Y$ where $X = f(A, B, C)$ and $Y = g(A, B, C)$. $A, B, C$ are 3 independent random variables and the functions $f, g$ are known but rather expensive ...
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210 views

Difference between identity and diagonal covariance matrices

thanks in advance for the help. Suppose I am training a linear model. What are the conceptual differences between using a diagonal covariance matrix and the identity? It is clear to me that the ...
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1answer
27 views

Covariance matrix computed based on a covariance function

I am reading Chapter 4 of Gaussian Processes for Machine Learning. It says that a matrix $K$ whose entries are computed as $k_{ij} = k(x_i, x_j)$ where $k$ is a covariance function is a positive ...
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1answer
150 views

Constraints on correlation coefficients of multiple random variables

I am looking for a generalization of Correlation between three variables question for more than three variables. It is stated in one of the answers there that, for three variables with pairwise ...
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2answers
826 views

How to find relation between 2 numbers

I have been practicing programming for many months now and what I found difficult is not about solving problem. But it is how to find the "how to solve problem" to make computer solves that for me! ...
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2answers
52 views

Correlation of random variables with joint PDF proportional to $x^{a-1}y^{b-1}(1-x-y)^{c-1} $

The random variables $X$ and $Y$ have joint PDF $$f(x,y)= \frac{\Gamma(a+b+c)}{\Gamma(a)\Gamma(b)\Gamma(c)}x^{a-1}y^{b-1}(1-x-y)^{c-1} $$ where $0 \leq x \leq 1 , 0 \leq y \leq 1, x+y < 1 $ where $...
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232 views

Checking hand manipulations of matrices

Beginning with a 4*3 matrix: 5 4 -1 2 3 -3 3 4 -4 1 3 -2 I have to perform four manipulations on it, which I did by hand. I wanted to ask if my thinking and/or ...
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2answers
207 views

What is the intuitive meaning of uncorrelated?

I was going through the derivation of the Kalman filter and it mentions that since noise (v) is uncorrelated to the state (x) and the state estimate (xbar), the following quantity is zero: E((x - ...
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2answers
70 views

Probability of observing a false correlation and confidence limits

In oil and gas exploration/development it is common to use acustic impedance derived from reflection seismic surveys to predict the porosity measured in wells drilled in the reservoir. I often use ...
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1answer
20 views

Re-calculating Value of $100 in Each State by Specific State

I'm using this Tax Foundations graphic for data. How would I re-calculate each state based on a specific state? For example, what if I wanted to base the control state on Missouri, which is $113.51. ...
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1answer
59 views

Determine Patterns

I have some weather data that I would like to analyze. I have about a millions rows of data, and each row has about 100 attribute values. Each attribute value represents some measurement (i.e., ...
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2answers
261 views

Autocorrelation function and power spectral density

I want to get the autocorrelation function of the power spectral density of the wind. This function is defined by: $$s(\omega)=\frac{c_1}{(1+1.5 \times c_2\omega)^{5/3}}$$ $c_1$ and $c_2$ are ...
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1answer
120 views

Regression with Mean, Standard Deviation, Range and Correlation

A research team collected data on students in a statistics course. Their dependent variable was the student’s score on the final examination, which ranged from 200 to 800 points. The observed average ...
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1answer
282 views

Weighing correlation by sample size

I'm a scholar in the humanities trying to not be a complete idiot about statistics. I have a problem relevant to some philological articles I'm writing. To avoid introducing the obscure technicalities ...
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1answer
66 views

Sample Size for Correlation Testing

A research team wishes to test the null hypothesis: $H_0, r=0$ at $\alpha = 0.025$ against the alternative: $H_1, r>0$ using Fisher’s transformation of the Pearson product moment correlation ...
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1answer
33 views

Work out if the relationship between 2 datasets is constant

I have 2 one-dimensional datasets, let's call them a and b. I want to know the correlation between ...
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1answer
301 views

What test should I use to statistically compare two intraclass correlation coefficients (ICC)?

I need to compare two generalizability (G) coefficients for data that are from two separate populations. G coefficients are a type of intraclass correlation coefficient (ICC). The literature on ...
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0answers
53 views

Correlation Statistic with a neutral score

I'm trying to develop a correlation statistic to compare lines/random variables (represented as a series of points). I want the correlation value to convey little information if one the lines is a ...
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1answer
132 views

Summing dependent random variables with unknown joint cdf

Suppose that X_1, X_2,... X_5000 are discrete and dependent non-identically distributed random variables, whose marginal distributions are known, but whose joint distribution is not known. Is there ...
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1answer
46 views

correlation estimator variance

Consider I have realisations of two random variables $X$ and $Y$ and I estimate their correlation thanks to the classic formula : $$\rho=\frac{\sum_{i=1}^{n}{x_iy_i}-\sum_{i=1}^{n}x_i\sum_{i=1}^{n}...
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1answer
93 views

correlation matching audio files peak position

if i cross correlate an audio file with itself (which means there is no delay or difference), the peak position should be a zero or at the center of the cross correlation? that is really confusing ...
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1answer
77 views

Convolution module

I am trying to apply GCC-PHAT algorithm here to process audio files and find delay between them. Im coding using Android and Java with the help of this library, and comparing the results with Matlab. ...
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240 views

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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1answer
315 views

Correlation coefficient of i.i.d variables

Let $X_1, X_2, X_3, ...$ be i.i.d variables, and for every $i$ $X_i$ has variance. Define $S_k=\sum_{i=1}^{k}X_i$. Calculate $\rho(S_m,S_n)$ for $m\leq n$. Well, I know it should be $\sqrt{ m/n }$, ...
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2answers
184 views

Finding the correlation coefficient of ordered statistics

I am working on the following problem. Let $$X_{(1)}, \ldots ,X_{(n)}$$ be the order statistics from the uniform distribution of $[0,1]$. Find the coefficient correlation of $X_{(1)}$ and $X_{(n)}...
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1answer
74 views

How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$?

Suppose we have a time series $X_t$ s.t. $X_t \sim^{iid} (0,1)$. How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$? Or, I guess, if $X,Y\sim^...
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2answers
171 views

Correlation coefficient calculation

Why do we remove of the mean of the data while calculating the correlation coefficient value of bivariate data in statistics? DotProduct/ProductOfLengthOfVectors should always give anyway a ...
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0answers
47 views

Multiple correlation

My question is very elementary. I need to implement a software that computes the multiple correlation of a set of datas that I have. In order to do that, I need would like to find the generic formula ...