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Questions tagged [correlation]

For questions about correlation of two random variables. Correlation is a statistical technique that can show whether and how strongly pairs of variables are related. Use it with [tag: random-variables] and [tag: probability].

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How to prove this relation for Kendall's distribution function (or Kendall's measure)

Kendall Distribution Function (Nelsen, 2006, p. 163) Or Kendall Measure (Salvadori et al., 2007, p. 148) Or Kendall Function (Joe, 2014, pp. 419–422) is the cumulative distribution function (CDF) of ...
khoshmard's user avatar
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2 answers
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Autocorrelation of Binary Random Variable

Let $X_i$ be a random variable that takes either the value $+1$ or $-1$. Suppose that if the previous value $X_{i-1}$ is $x\in \left\{ -1, 1\right\}$, then $X_i$ also takes value $x$ with probability $...
CRYPTONEWBIE's user avatar
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1 answer
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Is joint normal distribution flexible in correlation?

Suppose random variables $X$, $Y$, and $Z$ follow joint normal distribution. Conditional on $X$, we can calculate the correlation coefficient of $Y$ and $Z$. Is it always the case that this ...
Ypbor's user avatar
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Correlation Best Suited for Small Dataset? [migrated]

I have an x and a y that I would like to find the correlation of to learn more about their relationship. Unfortunately, I only have 10 points. Can I in good faith use the Pearson correlation ...
Camellia99's user avatar
2 votes
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Conditional Expectation for a multivariate normal distribution

I have $n$ random variables, $X_1, ..., X_n$, with covariance matrix, $\mathbf{\Sigma}^{n\times n}$ and I am trying to calculate $\mathbb{E}(X_1 | X_2 = x_1, ..., X_n = x_n)$. I am aware of the ...
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Removing Independent Variable Uncorrelated with All Other Variables in Linear Regression

I've been looking at Wooldrige's Introductory Econometrics and came across the following section related to omitted variables in multiple regression here The section essentially says that if an ...
Andrew Cheng's user avatar
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How to determine the correlation between a discrete variable and a continuous variable?

I have 9 conditions (discrete variables) with each having only two states: occurrence (corresponding to 1) and non-occurrence (corresponding to 0), resulting in a total of approximately 511 ...
Frontier_Setter's user avatar
2 votes
1 answer
54 views

Probability of coin flip given forecasts

Suppose you have a coin that flips $H$ or $T$ with some unknown probability. You also have access to two devices, $A$ and $B$, where $A$ correctly predicts the outcome of the coin with $p = 0.7$ and $...
shrizzy's user avatar
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What is the expectation of $R^2$ for for iid $y, x_1, ... x_k \sim N(0,1)$

$Y, X_1, ...., X_k$ are all iid $N(0,1)$ with $n$ samples. I can't make any progress on this problem... I don't even know an approximation but from simulation it seems to be $k/n$. What is the ...
user2330624's user avatar
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Formula for jointly correlated variables

I've always seen the following expression for normal correlated variables which is considered quite basic: \begin{equation} A=\rho Z+\sqrt{1-\rho^2}\epsilon \end{equation} I understand that it follows ...
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Proportion of variance in a linear regression model with a covaring predictor

Given a model: \begin{align}Y_{i}=Z_{i}*\beta * X_{i} + Z_{i}\tag{Eq. 1}&\end{align} I am interested in a closed formula for the proportion of variance explained by the predictor variable $X$, ...
CafféSospeso's user avatar
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Correlation vs Regression for a simple task

Hello everyone and thank you for taking the time with my issue! I want to apologize in advance if my question would've fit better on stack exchange, but I decided that the question is more related to ...
nicaaa's user avatar
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Why is the autocorrelation of an uncorrelated random noise process the dirac delta distribution?

I am reading Stochastic Methods by Gardiner and in the beginning of chapter 4 he motivates the rigorous interpretation of a Stochastic Differential equation by describing the properties of a "...
Mashe Burnedead's user avatar
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Which Test to Find a Correlation Between Two Selections.

I have a dataset of about 22 000 items, from which two processes have selected a small subset. I want to know if there is a correlation between the two selections, and I wonder which statistical test ...
Rafa's user avatar
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Is there a relationship between these two integrals?

I’m trying to find a (hopefully simple) relationship between these two integrals. Expressing the second integral in terms of the first one would be good. $\displaystyle \int_0^a f(x) (1-f(x)) dx$ and $...
EPH's user avatar
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$\rho_{X,Y}=1$ implies $Y-E[Y]$ is linear function of $X-E[X]$ with coefficient equals to $\frac{\sigma_Y}{\sigma_X}$

$\begin{align*} &\rho=1 \\ \implies &E\big((X-E(X))(Y-E(Y))\big)=\sqrt{E((X-E(X))^2)}\sqrt{E((Y-E(Y))^2)} \\ \implies & Y-E(Y)=a(X-E(X)) \end{align*}$ My hypothesis is that if $\rho=1$, ...
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Heat equation with Dirichlet boundary condition on a sole point

so I am working with the heat equation: $\frac{\partial G}{\partial t}(r,t) = D \Delta G(r,t)$ subject to the initial condition $G(r>0,t=0) = 0$ and the boundary condition $G(0,t>0) = 1$. I have ...
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If $X(t)$ is Ergodic, what about $X^2(t)$?

Given: ${X(t)}$ is W.S.S, Gauss with expected value $=0$, which has $R_{XX}(\tau)$ So $C_{XX}(\tau)=R_{XX}(\tau)$ $\int_0^{\infty}R_{XX}\left(\tau\right)d\tau<\infty$, since $X(t)$ is Ergodic. ...
Analysis_Complex_Study's user avatar
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Expected and Variance of the Squared Sample Correlation

I would like to obtain the expectation and variance of the squared sample correlation ($\operatorname{E}(R_{lk}^2)$ and $V(R_{lk}^2)$) between two random variables $l$ and $k$ following a bivariate ...
CafféSospeso's user avatar
0 votes
1 answer
31 views

strange bound on correlation for symmetric pdf

I am puzzled by a rather simple fact: The correlation of a symmetric multivariate pdf seems to be bound from below (increasingly strong with the number of dimensions). That seems unlikely to me. But I ...
zufall's user avatar
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If X and Y are uncorrelated, i.e., $Cov(X,Y) = 0$, then $E[Y | X] = c$ for some $c \in \mathbb{R}$.

It is given that X and Y are random variables on a common probability space $(\Omega, \mathcal{A}, P)$. Prove or disprove the following statement: If X and Y are uncorrelated, i.e., $Cov(X,Y) = 0$, ...
clementine1001's user avatar
2 votes
0 answers
56 views

Variable change in Copula for the joint pdf of correlated random variables

Let $f_{X,Y}(x,y)$ be the joint probability density of correlated random variables $X$ and $Y$ based on a Copula $C$ (Gaussian in my case) where $f_X(x)$ and $f_Y(y)$ are the marginal probability ...
Yakari Dubois's user avatar
2 votes
2 answers
89 views

How to prove $\int_{-\infty}^{+\infty}\langle f(t)f(t+s)\rangle\cos(a s)ds=a^2\int_{-\infty}^{+\infty}\langle\dot{f}(t)\dot{f}(t+s)\rangle\cos(as)ds$?

Denoting the time derivative of $f$ by by $\dot{f}$, I want to prove equation (2.96) given here: $$\int_{-\infty}^{+\infty} \langle f(t) f(t+s) \rangle \cos(a s) ds = a^2 ~\int_{-\infty}^{+\infty} \...
Mike's user avatar
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Where did the sin(kx)/(kx) term in the two-point correlation function $\xi(x)$ come from?

In my cosmology textbook, there is a derivation of the two-point correlation function (assuming statistically homogeneous and isotropic fluctuations). I follow the derivation up to equation (7.8a). ...
Hypatia of Alexandria's user avatar
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Is this particular function a positive semidefinite kernel function?

Curious if there is a proof or counter for whether the following function is a positive semidefinite kernel function. $K(x,y) = \max\left(b_0 - b_1 \frac{|x-y|}{x+y}, 0\right)$ with $x > 0, y > ...
pandasdataframe's user avatar
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How to construct a maximum-information embedding of sampled objects using a binary function?

This feels like a very specific problem, but I hope there already is a method to achieve what I want. There is a random process from which I can draw samples of non-numerical, variable sized objects (...
fazekaszs's user avatar
  • 153
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1 answer
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Sum of autocorrelation coefficients

This is a follow-up to this thread: Proof that sum over autocorrelations is -1/2 I am posting a new thread as that was posted 6 years ago. In that threat the stackexhange author (Kuhlambo) lists some ...
Bazool's user avatar
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1 vote
1 answer
29 views

Correlation of sum

What is Corr(X+Y, Z) in terms of Corr(X, Z) and Corr(Y, Z)? If there is no direct formula, is there any relationship in terms of > or <? Also, the same thing for rank correlation and Kendall's ...
Anonymous's user avatar
1 vote
0 answers
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What does it really mean to take correlation between time series? [closed]

I have a conceptual problem when we extend the correlation to time series. I understand probability and statistics as a two way route. Either I begin from a random variable (r.v.) $X$ and sample from ...
Curious student's user avatar
-1 votes
1 answer
30 views

Show that rank correlation lies between -1 and 1

Show that Spearman's rank correlation r lies between -1 and 1. where $r = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}$
Shub's user avatar
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0 answers
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Two-time correlation function from probability distribution

Given a continuous stochastic process $x_t$ defined by the following Langevin equation \begin{equation} d x_t = dB_t +F(x_t)dt \end{equation} where $dB_t$ is a Wiener increment and $F(x_t)$ is a ...
J.Agusti's user avatar
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1 vote
1 answer
58 views

$E[\exp(-sX)\exp(-sY)]$ for two identically distributed, but correlated random variables X and Y.

I am trying to figure out the following problem. I am trying to evaluate the expectation: $E[\exp(-sX)\exp(-sY)]$, where $X$ and $Y$ are identically distributed, but correlated random variables, hence,...
SecretKeeper's user avatar
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0 answers
27 views

Is it possible to reverse the Pearsonr?

Given a Pearson correlation coefficient (PCC) and a list of values (target), can I generate another list of values with roughly the same PCC as the targeting list? For example, given a list of values ...
sensationti's user avatar
0 votes
1 answer
30 views

Intuitive understanding of multipying a matrix by a vector and its hermitian transpose

I keep seeing $$\overline{a}^H \text{ } \overline{R}_{xx} \text{ } \overline{a} = \text{single value}$$ Where $\overline{a}$ is a mx1 vector and $R_{xx}$ is a mxm autocorrelation matrix of another ...
Villere_DSP's user avatar
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0 answers
13 views

Using graphs to quantify the structure/pattern or correlation among the elements of supposedly random matrix

Let's say I have a supposedly random real symmetric matrix. How to use graphs to quantitatively (with a numerical focus) examine any structure/pattern or correlation among its elements ?
Snpr_Physics's user avatar
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0 answers
10 views

What formula should I use for 1D autocorrelation?

I obtained a list of $r^2_{end-to-end}$ from a Monte Carlo simulation of polymer movement. ...
user366312's user avatar
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1 vote
0 answers
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Two Gaussian which are not jointly Gaussians but not dependent

Let $X \sim N(0,1)$ and B such that $P(B=1)=P(B=-1)=1/2$, two independent variables. Define - Y=BX. Then Y is also Gaussian since - $$F_{Y}(y)=P(Y\leq y)=P(BX \leq y)=P(B=1)P(BX \leq y|B=1)+P(B=-1)P(...
Yar Sha's user avatar
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0 answers
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Transformations to solve question surrounding correlation of vectors

I am trying to solve part (b) of this question, but I am having trouble with something. With some help, I have been able to solve the following: $$ \begin{align*} \max_{a \neq 0, b \neq 0}\...
Lucius Aelius Seianus's user avatar
0 votes
1 answer
65 views

Using transformations to solve question surrounding correlation of vectors

While studying statistical inference and data-analysis, I came across this question. I have been able to show part (a) of this question already myself, but I don't know how to solve part (b). Can ...
Lucius Aelius Seianus's user avatar
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0 answers
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How to determine the significance of a difference in a pearson correlation?

I'm developing some methods for processing images. I have applied seven methods (M1 to M2) in 10000 images and measured the effects of each method in each image according to two performance measures. ...
Zaratruta's user avatar
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1 vote
1 answer
60 views

Correlation and independence of two random variables

Let $X\sim N(0,1)$ and $Y=(-1)^J \cdot X$, where $P(J=1)=\frac{1}{2}=P(J=0)$. Furthermore X and J are independent. I already showed that $Y\sim N(0,1)$ by calculating the cdf of Y. Now I want to show ...
Blue2001's user avatar
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1 answer
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Finding the covariance and correlation of two random variables

Let $X$ be a random variable that has a standard uniform distribution $U(0,1)$, let $Y = X^k$, $k > 0$. I have performed the random variable transformation receiving $g(y) = \frac{1}{k}y^{\frac{1}{...
milosz7's user avatar
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1 answer
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Prove two random variables are negatively correlated

Given a raw estimator \begin{equation} I=I(X>b)= \begin{cases}1, \text{ if } X > b\\ 0, \text{ otherwise } \end{cases} \end{equation} for some random variable $X$ and some constant $b$ in the ...
Analysis Rookie's user avatar
0 votes
1 answer
77 views

2 Brownian Motions with Non Zero Correlation and NOT jointly normal?

Is it possible for 2 Brownian Motions to have non-zero correlation without being jointly normal? I'm a bit confused by the question. I just assumed we always talk of multiple Brownian Motions as being ...
jmac's user avatar
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1 vote
0 answers
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Combinations of random vectors to represent a set of points

I have following problem and can't find anything in the direction I am thinking about: I have a given price index my company collects for the last 20 quarters or so and have a tool that shows me the ...
durst's user avatar
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1 vote
1 answer
70 views

Why is $xx'$ singular but ${\Bbb E}[xx']$ not (necessarily)?

Assume $x$ are random vectors and $x'$ denotes the transpose of vector $x$. In Hansen's Econometrics an assumption of the Linear Predictor model is that ${\Bbb E} [xx']$ is positive definite. I get ...
ArOk's user avatar
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1 answer
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$Y_n = V_n + a V_{n-1}$, $V_n$ iid with $\mathbb{E}[V_n] = 0$ and variance $\sigma_V^2$, find autocorrelation for $Y_n$

I’m struggling with how to express the solution to the question in terms of the variance. this is the solution$$\mathbb{E}(Y_n Y_{n+m}) = \sigma_V^2 \cdot [(1+n(n+m))\delta(m) +n\delta(m+1)+ (n+m)\...
kal_elk122's user avatar
2 votes
1 answer
55 views

Using Orthogonal Decomposition of Bivariate Normally Distributed X,Y Reveals Different Result Than Calculating Variance Directly

This is the exercise description: 6.5.4. Suppose X and Y are standard normal variables. Find an expression for P(X + 2Y ≤ 3) in terms of the standard normal distribution function Φ, (a) in case X and ...
BurgerMan's user avatar
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1 answer
57 views

Understanding correlation in the context of a time series, simulation and brownian motion

I have the doubt of calulating and meaning of correlation. I know it is from my incapacity to grasp a concept, specially regarding time series but would appreciate any comments on it. I think I ...
Curious student's user avatar
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0 answers
42 views

What is the intuitive difference between covariance and the product of the standard deviations of bivariate data?

Can someone help me understand what the difference is—intuitively—between these concepts? Here's how I currently think of these things. The covariance, represented by: ... is essentially a mean area, ...
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