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

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Solving for the covariance of a joint pdf

Let X and Y have a joint pdf given by $f_{x,y}(x,y) = \begin{cases} 1 & \text{if } 0<y<1,\text{ } y-1<x<1-y \\ 0 & \text{otherwise} \end{cases}$. (a) Find Cov(X,Y) and ...
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0answers
88 views

Expectation of random variables

a) Show that $E\{X-E(X)\} = 0$ for any random variable $X$. b) Use the result in part (a) and the following equation to show that if two random variables are independent then they are uncorrelated, If ...
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1answer
79 views

A question about Pearson correlation coefficient

Suppose that we have two vectors $x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n)$ is the following correct about their Pearson correlation coefficient? $\operatorname{corr}(x,y)=\operatorname{corr}(x+a,y+b)$ ...
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1answer
36 views

Curve Fitting and Multiple Experiments

Say I do an an experiment 5 times, each of which gives you a list of data points. Do I fit a curve to each one separately and then average the parameters and their uncertainties? Or do I take the ...
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0answers
77 views

When does convergence in distributions inply convergence in covariance?

Good Morning. Let $(X_n)_n$ and $(Y_n)_n$ be sequences of random variables converging in distribution respectively to $X$ e $Y$. Suppose $X_n,Y_n$ are equally distributed but dependent for all $n$, ...
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38 views

Finding correlation of data with potentially hidden time lags

Let's say I have few independent variables plus multiple observables that I monitor over time for a system. I'd like to find out if there is any correlation between the observables and any of the ...
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1answer
55 views

Multivariate Gaussian density from singular covariance

I have a multi-dimensional (~600dim) sample from which I determine its covariance matrix. The determinate of the covariance is 0. The sample does not show strong correlations when plotting 2 ...
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1answer
185 views

Finding the joint distribution of a random process with memory

I'm modeling a digital system as a random process and attempting to solve for the autocorrelation in order to arrive at the power spectral density of the process. The system is as follows: At any ...
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0answers
80 views

Expected value, correlation, and indepence.

I need help with a problem. Supposed x, y, and z are events in F (algebra of sets) in a probability space (universal set, F (algebra of sets), P). Define two random variables: a(omega) = ...
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1answer
87 views

Autocovariance of a given stochastic process?

I need to find the autocovariance $C_{YY}(t,s)$ of the stochastic process $Y(t) = t^2 X(t) -2X'(t)$ where $C_{XX}(t,s) = e^{-t^2 -s^2}$ is given. Using known properties I can calculate the ...
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2answers
123 views

Why is this convolution true?

I am a little puzzled by how the following summation has been written as a convolution, with one of the inputs reversed in time. Consider the following sum on the LHS, and the convolution on the RHS. ...
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0answers
42 views

Question on change of variables during convolution/correlation

I am trying to understand how the following two statements are equivalent: $$ \sum_{l=-\infty}^{\infty} h^*[l] \ R_{xx}[m+l] = \sum_{i=-\infty}^{\infty} h^*[i-m] \ R_{xx}[i] $$ I get that we made ...
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0answers
49 views

A vector with fixed correlation with existing vector, is it always possible?

Suppose we have a known vector $X$ in $R^n$, and for any vector $Y$ in $R^n$, we impose on it the restriction that it must have a fixed correlation coefficient $r$ with $X$: \begin{align*} ...
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1answer
143 views

Correlation of sums of correlated variables

I'm trying to work out an expression for a correlation of the weighted sums of two r.v.'s with a third r.v. To be precise, I have a trivariate normal distribution: $$\{X,Y,Z\}\approx ...
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1answer
415 views

Proving $Y = aX + b$ given correlation coefficient $|\rho(X, Y)| = 1$

With correlation coefficient defined as: $$\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X)}\sqrt{\text{Var}(Y)}}$$ can you help me prove $$|\rho(X, Y)| = 1 \implies Y = aX + b$$
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1answer
205 views

Mean density of the nontrivial zeros of the Riemann zeta function

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ...
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1answer
621 views

Asymptotic correlation between sample mean and sample median

Suppose $X_1,X_2,\cdots$ are i.i.d. $N(\mu,1)$. Show that the asymptotic correlation between sample mean and sample median (after suitably centering and renormalization) is $\sqrt{\frac{2}{\pi}}$.
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143 views

Probability and correlation function, interpretation of a result

My question is originated from the paper ...
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1answer
97 views

Finding parameters for curve fitting

I have 500 observed data of variable $ x $ and corresponding $ y $. The functional model is where Is it possible to find suitable constants $ A , B $ ,$ \alpha , \beta $ so that the observed ...
5
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1answer
8k views

Generate Correlated Normal Random Variables

I know that for the $2$-dimensional case: given a correlation $\rho$ you can generate the first and second values, $ X_1 $ and $X_2$, from the standard normal distribution. Then from there make $X_3$ ...
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0answers
33 views

Correlation between properties

I have a table in a database where each record includes the ID of a person and the ID of a property of that person. Each person may have more than one property. Which is the statistical instrument ...
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1answer
259 views

Related rates, where do I start?

A revolving searchlight, which is $100$ m from the nearest point on a straight highway, casts a horizontal beam along a highway. The beam leaves the spotlight at an angle of $\frac{π}{16}$ rad and ...
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36 views

Finding correlations between many unknown functions.

Given an arbitrarily large number of black-box functions of one variable, is it possible to produce expressions that approximate their relationships to each other over their shared domain? Is it ...
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2answers
1k views

help understanding step in derivation of correlation coefficient

I'm looking to understand the starred step in the derivation below (also, if someone could help with the LaTex alignment, I'd appreciate it). The regression line is $y= b_0 + b_1 x$, where $b_0$ and ...
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0answers
113 views

Correlation function of Brownian motion. What am I doing wrong?

Can anyone tell me where I am going wrong here? (I am leaving out any random fluctuation forcings, because I don't think they are relevant to my problem.) 1: $\displaystyle \frac{dv(t)}{dt}=-\eta ...
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0answers
82 views

Calculation of conditional joint probability given certain conditionals for data which aren't independent

The context of this problem is the estimation of the distribution of a parameter $v$ given sets of data $A$ and $B$, where $A$ and $B$ are not independent. Suppose I know $P(v | A)$ and $P(v | B)$. ...
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1answer
96 views

correlation of product with its normally distributed factors

If x and y are normally dist. with standard deviation of 10%, and they are independent, then their product X.Y is 71% correlated with Y (or X). I can show this empirically, but how to I prove it in ...
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2answers
206 views

Trying to understand correlation and independence geometrically

I am trying to understand correlation and independence of two random variables geometrically, but found it difficult to grasp the intuition and to explain it rigorously. First, given two uniformly ...
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0answers
63 views

Dimension free Concentration bounds for Martingales

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
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0answers
111 views

Finding the empirical correlation from a covariance matrix

I have this covariance matrix with five variables $X_1$ through $X_5$ in that order. ...
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1answer
356 views

Random walk serial correlation

Given a model $$Y_t =b_0 + b_1 \cdot X_t + b_2 \cdot Z_t + e_t,$$ where the error term $e_t$ follows a random walk form of serial correlation $e_t = e_{t-1} + u_t$. Further assume $u_t$ has zero mean ...
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1answer
545 views

Inequality concerning the pairwise correlation coefficients of three random variables

I was asked to prove: The correlation coefficients, $\rho_{12}$, $\rho_{23}$, $\rho_{13}$ between three random variables $X_1$, $X_2$, $X_3$ obey ...
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1answer
153 views

Covariance and Correlation

Suppose there were m married couples, but d of these 2m people have died. Regard the d deaths as striking the 2m people at random. Let X be the number of surviving couples. Find: a) E(X) b) Var(X) ...
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1answer
331 views

Correlation of Indicator Variables

Show that for indicator random variables $I_A$ and $I_B$ of Events $A$ and $B$: $Corr(I_A, I_B) = Corr(I_A^c, I_B^c) = -Corr(I_A, I_B^c) = -Corr(I_A^c, I_B)$ Deduce that if $A$ and $B$ are ...
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1answer
77 views

How can I mathematically show the similarity between these 3 plots?

I have 3 3D plots of field strength measured around an antenna. I want to calculate the mathematical similarity between the points of the field patterns. How can I do this? thanks
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1answer
42 views

What values to choose for correlation?

To work out correlation I'm using the online calculator : http://easycalculation.com/statistics/correlation.php ...
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0answers
1k views

What do angle brackets ($\langle\rangle$ ) mean in mathematics/statistics (autocorrelations)?

Okay, so the logarithmic return on a stock is given by: $$r_τ (t) = \ln P(t+τ) - \ln P(t),$$ where τ is the interval of time. I have no problem calculating that. My question comes to the following ...
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1answer
163 views

Find $\operatorname{argmax}_x \operatorname{corr}(Ax, Bx)$ for vector $x$, matrices $A$ and $B$

This is similar to, but not the same as, canonical correlation: For $(n \times m)$ matrices $A$ and $B$, and unit vector $(m \times 1)$ $x$, is there a closed-form solution to maximize the correlation ...
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1answer
19 views

Correlation Based Filter

i found this paper. Im interesting in part 2.3 Feature Weighting. The correlation function is known from wikipedia and almoast clear ( i can write a function to calculate the value :) ) But now i ...
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1answer
184 views

Is the relation of having positive covariance well behaved with respect to taking the inverse?

Let $X$ and $Y$ be two random variables, $X$ strictly positive. Assume that Cov$(X,Y)>0$. Does this imply that Cov$(1/X, Y)<0$? I know that being positively correlated is not a transitive ...
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1answer
154 views

special matrix in terms of its covariance matrix

How can we find a matrix $S\in \mathcal{M}_{n,n}$ and $Z\in \mathcal{M}_{n,m}$ whose $n$ entries of the $i^{th}$ column $Z_i$ are correlated $Z_i \sim \mathcal{N}(0,S)$ where $S \in \mathcal{M}_{n,n}$ ...
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2answers
107 views

Covariance$(X,Y) \geq 0$ if $X,Y \geq 0$?

I was wondering if you can say something about the covariance of two positive variables $X$ and $Y$?
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1answer
183 views

How to correlate the timestamps of 2 systems?

Whenever I've done (simple) correlation in the past, I've always had 2 sets of data that had "connected" axes: ...
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0answers
111 views

Is the correlation function convex or not?

Suppose the function for statistical correlation is a non linear constraint in a non linear programming model: $$ \frac{\sum_{t=1}^T (p_t - \bar{p})(R_t - \bar{R})}{\sqrt{\sum_{t=1}^T (p_t - ...
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1answer
84 views

What is the correlation function in multivariable/vectoral case?

I know that the correlation function between random variables $X$ and $Y$ is defined as $$ \rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over ...
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2answers
2k views

What is a direct correlation?

I have two contrary definitions of for the direct correlation between two variables $X$ and $Y$ Their correlation coefficient is close to $1$. There is a direct causal relationship between the ...
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129 views

Does $0$ correlation imply independence for marginally normal distributions?

Assume $X \sim \mathcal N(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal N(\mu_2, \sigma_2^2)$. If $\rho_{X,Y} = 0$ then $X \bot Y$. Can someone give a hint why this is true ?
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0answers
48 views

Sorting Vectors based on their correlation

This problem [question]: Sort vectors according to their distance between them is about sorting vectors based on the distance between them. What about sorting vectors based on the correlation ...
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1answer
132 views

The science of pearson product moment correlation coefficient

I need to compare two sound signals for similarity, I took cross-correlation of both the signals and I got a cross-correlation signal, now I intend to use pearson correlation coeff formula to get the ...
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1answer
216 views

Pearson Correlation Coefficient Interpretation

Let $X=(1,2,3,...,20)$. Suppose that $Y=(y_1,y_2,...,y_{20})$ with $y_i=x_i^2$ and $Z=(z_1,z_2,...,z_{20})$ with $z_i=e^{x_i}$. Pearson correlation coefficient is defined by formula \begin{equation} ...