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

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Correlation of two random variables

A random sample of $100$ variables is given. Each of them is independent and identically distributed with $N(0,1)$. What is the correlation between sum of $98$ variables and sum of $100$ variables?
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47 views

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}_i) = 0$

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}) = 0$ in the image below (third and fourth line of the proof!). Why?
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21 views

Min/Max variance of 3 correlated variables

You have 3 different random variables (assets) all with exactly the same variance. What is the maximum and minimum variance of the 3 variables (assets) combined? Proposed ...
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39 views

testing correlation coefficient in a bivariate normal distribution

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 $(0,1)$ ...
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27 views

Random Variables, Variance and Coefficients

So, I'm trying to get my head around a question and I was wondering if you could help me. Given that the correlation coefficient $\rho$ for random variables $X$ and $Y$ is: $\;\rho_{\lower{0.5ex}{X,...
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22 views

explanation of correlation of stationary stochastic processes

I have some questions about correlation in stationary stochastic processes. I know that the expectation of a random variable is $E(x)=\int_{-\infty}^{+\infty} a ...
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28 views

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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1answer
47 views

Probability at least one, using correlation

I have a problem using the correlation in combination with the "at least one" probability. I have $P(A)=57\%$, and $P(B)=74\%$, and I calculated their correlation coefficient and it is $0.1557$. To ...
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1answer
23 views

How to find the percent variation in Y is explained by X?

I know that the r^2 value for the data is 0.9832. Is there a way to use that value to find the percent variation in Y is explained by X? Or do I need to use the data given to me?
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2answers
31 views

Uncorrelated and X given $Y = 0$

Is the following true or false? Suppose that $X$ and $Y$ are two discrete random variables defined on the same probability space. If $E[X] = E[Y] = 0$ and $E[X | Y=y] = 0$ for all $y\in Y$, then $X$ ...
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1answer
98 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 ...
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1answer
319 views

Finding the autocorrelation of a sine wave.

The autocorrelation of sin(t) is defined as $$\displaystyle \int_{-\infty}^{\infty} \sin(t+\tau)\sin(t)d\tau$$ I've tried using the Wiener-Khinchin theorem which says that $$Corr(g,g)\...
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75 views

Cauchy Schwarz inequality in Normalized Cross Correlation

I'm currently using a normalized cross correlation(NCC) for measure the degree of similarity between two image. Almost two week studying about how NCC is derived from Cauchy Schwarz inequality but ...
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1answer
46 views

Meaning of “White noise uncorrelated in time”?

On page 3 of these slides, it says: It’s often a good approximation to make the random force “white” and “uncorrelated in time”. And this equation just pops out of nowhere. I know white ...
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38 views

How many variables can be pairwise anticorrelated

I am working on a computational project involving analysis of data. Each item of data that I have has a few hundred attributes; I have several million items of data. The attributes are essentially ...
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1answer
11 views

Make an equation inversely relating a radius to a punctuation

Lets have a shooting target. (Sorry, can't give an example image since I'm on mobile data) It has several rings which represent the score. the outer ring is 1, and the inner one is 11. Lets assume ...
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3answers
180 views

Help with understanding point from Kahneman's book “Thinking Fast and Slow”

My question: How did Kahneman arrive at the 60% number in the last sentence ("60% of the pairs")? From Daniel Kahneman, Thinking Fast and Slow (Chapter 19, Illusion of Understanding): Update: from ...
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23 views

Correlation and entropy between stocks of the same index

My portofolio contains the stocks belonging to Nasdaq100 index. Initially, i found the entropy of closing prices between Friday's value and Monday's value,for all companies of the index, in order to ...
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68 views
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why do odd magic squares have the same rank as their size?

why do odd magic squares have the same rank as their size whats special about odd magic squares? and why do even magic squares alternate The results are below where n is the size and r is the ...
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1answer
53 views

Correlation between three variables

Here is my question. If A and B has a correlation of 0.8, B and C has a correlation of 0.6. What is the range of correlation between A and C?
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2answers
85 views

Why is $R^2=\rho^2$

Considering $y_i=\beta_1+\beta_2x_i+\epsilon_i$ $\bar y_i=\hat\beta_1+\hat\beta_2\bar x_i+\bar\epsilon_i$ a linear equation of least square used when it seems that there is a like between two data, $...
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21 views

Formula for the n'th order correlator $\langle \Gamma(t_1)\Gamma(t_2)…\Gamma(t_n)\rangle$ of Gaussian white-noise?

Is there a closed form formula for the n'th order correlator $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\langle \Gamma(t_1)\Gamma(t_2)...\Gamma(t_n)\rangle$, of Gaussian white-noise $\Gamma(...
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54 views

Show $X_1$ and $X_2$ are negatively correlated

Consider $n$ independent tosses of a die. Each toss has probability $p_i$ of resulting in $i$. Let $X_i$ be the number of tosses that result in $i$. Show that $X_1$ and $X_2$ are negatively correlated....
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21 views

Correlation matrix

I have this example of a correlation matrix in my notes: $$R = E[XX^T]$$ $$n = 3, E[X_i X_{j}^*] = 2^{|i-j|}$$ $$R = \left[ \begin{matrix} 1 & 0.5 & 0.25 \\ 0.5 & 1 & ...
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22 views

Is the sum of all cross-correlation samples representative of target existence likelihood?

Answers to this question take the peak in the cross correlation as the measure to the likelihood of the trigger signal exist in the received signal - this is pretty much text book. My question is ...
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50 views

What is the correlation of two normal distributions with equal mean and known relation between variance

If I take the sample mean of a scaled $\chi$ distribution of $N$ samples, the distributions of these means should lead to a normal distribution $\mathcal{N}(\mu,\sigma)$. This procedure is repeated $M$...
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2answers
144 views

Conditions for the convergence of two sorted vectors of samples

Let $X$ be a random variable and $X_1, X_2, \ldots, X_n$ be a sample of size $n$ and $X_{(1)},X_{(2)},\ldots,X_{(n)}$ the corresponding order statistics, which are obtained by sorting the values $X_1, ...
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53 views

Conditional Expectation of random sum of independent random variables (when $N$ and $X_i$ are dependent)

In this question, $Y=X_1+X_2+\dots+X_N$ where $X_1,X_2,\dots,N$ are jointly independent random variables, $X_1,X_2 ...$ identically distributed continuous random variables with finite expectation, ...
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1answer
39 views

Given three sets of data A, B and C. Can A correlate with C, and C with B, but not A with B

Here are three sets of data: A B C 0 0 4.84 2 0 4.28 1 1.73 2.74 You can check that the (product moment) correlation coefficient between A ...
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45 views

What is the correlation between these two vectors?

Assume we have constant positive numbers $k$ and $N$ and a vector of the following form $$ \tag 1 a(\theta_1) = \frac{1}{\sqrt{N}}[ 1, e^{jk \sin(\theta_1)},e^{jk \sin(\theta_1)}, \cdots, e^{j(N-1)k\...
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1answer
24 views

Covariance and correlation, and how are they related?

I get that corellation is the covariance divided by the multiplie variance of the two, uh, things. What i don't get is why they are divided by the multiplied variance, and why that limits the value ...
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40 views

Is there any measure for detecting quadratic or cubic relationships?

Correlation coefficients is a useful measure for detecting linear relationships. Is there any measure for detecting linear relationship between one dependent variable and more than one independent ...
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63 views

Correlation function of an asymptotically stationary AR process

I have a great confusion with the autocorrelation function of an AR process. Its derivation usually follows in this way (Haykin, 2007): The difference equation for an AR(M) process, $u(n)$, is \...
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1answer
24 views

Correlation coefficient and orthogonality

In the book Matrix Analysis and Applied Linear Algebra, the author describes the coefficient of linear correlation as $$\frac{(x-\mu_xe)^T(y-\mu_ye)}{||x-\mu_xe||\cdot||y-\mu_ye||}$$ where $x,y\in \...
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1answer
134 views

Linear correlation in 3D?

What's the name of a statistical method used to determine the goodness of fit of a series of points in 3D space that are to be fitted on a regression line ? I can calculate a regression line and the ...
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23 views

Generating random variables with specific higher order correlations

I would like to generate a series of random variables {$X_1,X_2,...$} with the following properties: 1) $\langle X_i\rangle=1$ for all i. 2) $\langle X_i \cdot X_j \cdot X_k \cdot ...\rangle=0$ ...
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2answers
25 views

Identifying modules in a correlation matrix

I am a biologist with some maths background... but not enough knowledge to solve this problem, so I would be really grateful if someone could help (and explain it at a level that a biologist might ...
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18 views

Autocorrelation of a signal known only on an interval of finite size

Let's consider we have a continuous random signal ${ t \in ] - \infty \,;\, + \infty [ \mapsto b (t)}$. We assume this signal to be stationary, so that when ensemble-averaged, one may introduce the ...
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48 views

Limit of Covariance and Correlation of Random Walk

Let $S_n=X_1+...+X_n$, $n\geq1$ be a random walk, where $EX_k=\mu$ and $Var(X_k)=\sigma^2$, $0<\sigma^2<\infty$. a)Find the covariance $Cov(S_n,S_m)$ and the correlation coefficient $\rho(S_n,...
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1answer
51 views

Is there a simpler way to calculate correlation?

Let's consider that a variable y constructed from x $x_i ∈ \left\{1,3,5,7,8\right\}$ $f(x_i)=2x_i+1$ $y_i=f(x_i) + ε_i, ∀i∈ \left\{1;...;5\right\} $ where $ε_i$ is a identically ...
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0answers
21 views

Variance of estimating coefficients by correlating a sequence

I have a sequence $$ r[n] = a_1.t_1[n] + a_2.t_2[n] + a_3.t_3[n] + ... $$ where $t_1, t_2, t_3,...$ are uncorrelated, two-level (+A/-A), zero mean, pseudo-random sequences. To estimate $a_1$, $r[n]$...
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18 views

Why are the Real and Imaginary parts of a field in k-space uncorrelated?

I am in the process of generating a (real) Gaussian random field $\delta(\vec{x})$ from a given power spectrum $P(k)$. The way I define the power spectrum is, in Fourier space, $\left\langle \delta(\...
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23 views

Properties between Pearson correlation of two function and the correlations of their derivatives

For two functions, $f(t)$ and $g(t)$, I can calculate the Pearson correlation $Corr$. Does the correlation, $CorrD$, between the derivative functions, $f'(t)$ and $g'(t)$, has some properties with ...
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1answer
140 views

The relationship between diagonal entries and eigenvalues of a diagonalizable matrix

Let $\mathbf{C}$ be an $n\times n$ Hermitian matrix. Let $\dagger$ indicate a matrix conjugate-transpose. Let $\mathbf{V}\mathbf{D}\mathbf{V}^\dagger$ be the eigendecomposition of $\mathbf{C}$, where ...
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33 views

similarity between two ranked sequence

How can I measure similarity/distance between two sequences of ranked numbers/letters. The two sequences are of different length, and only have some elements in common? For example, if I have three ...
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0answers
30 views

Concentration bounds on Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(...
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1answer
61 views

If $X$ and $Y$ are Normally distributed with correlation $\rho$, can we say anything about $E[Y \mid X]?$

Let $X \sim N(0, 1)$ and $Y \sim N(0, 1)$ and $\mathbb E[XY]=\rho$. Can one say anything about the conditional expectation $\mathbb E[X \mid Y]$? In general, this clearly does not seem to work, ...
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1answer
29 views

inference of causality for binary variables

Let's say that a data set has N random binary variables Xi and we want to infer which of these variables have a causal relationship with X1. The following table would describe the data, where each ...