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

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6
votes
1answer
13k 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 $X_3$ ...
3
votes
3answers
5k views

Correlation between two linear sums of random variables

I understand how to create random variables with a prespecified correlational structure using a Cholsesky decomposition. But I would like to be able to solve the inverse problem: Given random ...
8
votes
1answer
736 views

Covariance, covariance operator, and covariance function

I am trying to get my head wrapped around this article in Wikipedia. The first definition given there is the covariance of a probability measure $\mathbf{P}$: $$\mathrm{Cov}(x, y) = \int_{H} \langle ...
3
votes
4answers
10k views

Correlation between three variables question

I was asked this question regarding correlation recently, and although it seems intuitive, I still haven't worked out the answer satisfactorily. I hope you can help me out with this seemingly simple ...
1
vote
1answer
72 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 ...
16
votes
2answers
17k views

Generating correlated random numbers: Why does Cholesky decomposition work?

Let's say I want to generate correlated random variables. I understand that I can use Cholesky decomposition of the correlation matrix to obtain the correlated values. If $C$ is the correlation ...
3
votes
3answers
675 views

Correlation between variables

I asked this question on stats SE but did not find a suitable answer so far. Maybe someone can help. Given n random variables x1,...,xn (one-dimensional). The following is known (corr() = Pearson ...
2
votes
2answers
14k views

Calculating the variance of the ratio of random variables

I want to calculate $\newcommand{\var}{\mathrm{var}}\var(X/Y)$. I know that the solution is $$\var(X) + \var(Y) - 2 \var(X) \var(Y) \mathrm{corr}(X,Y) \>,$$ but, how do I derive it from "common" ...
2
votes
2answers
176 views

Given X and Y are correlated and Y and Z are correlated what is the range of correlation between X and Z?

How can I calculate the range of correlation of two variables X and Z given I have the correlations of X and Y, and Y and Z? I've found a few resources around, namely this, but I'd like a research ...
6
votes
1answer
221 views

Correlations between neighboring Voronoi cells

For a sequence $X_1,X_2,X_3,\ldots$ of random variables, what it means to say $X_1$ is correlated with $X_2$ is unambiguous. It may be that the bigger $X_1$ is, the bigger $X_2$ is likely to be. If, ...
4
votes
3answers
1k views

Bounds on off-diagonal entries of a correlation matrix

Assume that all the entries of an $n \times n$ correlation matrix which are not on the main diagonal are equal to $q$. Find upper and lower bounds on the possible values of $q$. I know that the ...
2
votes
1answer
95 views

Wedge Product Formula For Sine. Analogous Formula Generalizing Cosine to Higher Dimensions?

So I was day dreaming about linear algebra today (in a class which had nothing to do with linear algebra), when I stumbled across an interesting relationship. I was thinking about how determinants are ...
2
votes
1answer
2k views

Maximum and minimum Correlation Coefficient

I have a question regarding the correlation coefficient. The inspiration is from a story where a student collected a set of $(X,Y)$ pairs, but lost the pairings. Hence, he is left with two sets of ...
1
vote
2answers
151 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 ?
1
vote
3answers
1k views

Determinant of a N symmetric square matrix with diagonal 1

What is the determinant of a symmetric $n \times n$ matrix with all diagonals be 1 and all others are $\rho$ (yes correlation matrix)? Anyone can tell me a method to work it out elegantly? Thanks!
3
votes
1answer
2k views

quadratic relationship

Detection of linear relationship is possible with correlation coefficient. If absolute value of correlation coefficient is 1, then the relationship is linear. Is there any way for detecting quadratic ...
3
votes
0answers
173 views

How to perform nonlinear regression with correlated errors?

I have a nonlinear least squares problem, but the errors are correlated. I could use R's nls function to do the regression if the errors were independent, but I don't know the right way to handle ...
2
votes
1answer
113 views

Simulate correlated $\chi^2$ distribution

I understand that when one have multiple independent variable that follows $N\sim(0,1)$, denoted as $A$ if we have a correlation matrix $R$, we can generate correlated variables $B$ that are normally ...
1
vote
2answers
333 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, ...
1
vote
1answer
233 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 ...
1
vote
0answers
125 views

Windowed Linear Correlation

$\DeclareMathOperator \Cov {Cov}$ $\DeclareMathOperator \Var {Var}$ $\DeclareMathOperator \E {E}$ Consider the following experiment: For $N\geq1$, consider $N$ black balls. Let us paint each black ...
1
vote
1answer
877 views

Expectation product of pairwise uncorrelated variables

Suppose I have three uncorrelated random variables $X, Y$ and $Z$ (discrete or continuous) such that $$\newcommand{\Cov}{\mathrm{Cov}}\Cov(X,Y)=0;\quad \Cov(Y,Z)=0;\quad \Cov(X,Z)=0 \tag{$\ast$}$$ I ...
0
votes
1answer
46 views

Reconciling different definitions of orthogonality

I want to establish about orthogonality in my mind. I knew the orthogonality of two functions $f$ and $g$ in interval T like the following: $$ \int_{<T>}f(t)g^*(t)~dt=0 \tag{1} $$ where $$ ...
0
votes
0answers
17 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 ...
0
votes
1answer
74 views

sufficient conditions for a stochastic process to be wide sense stationary

From the page Stationary process, I have the following definition: WSS random processes only require that 1st moment and autocovariance do not vary with respect to time and from the page ...
0
votes
1answer
156 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 $S \in \mathcal{M}_{n,n}$ ...
0
votes
1answer
129 views

correlation between two different variables

I am studying stochastic processes and found the next problem: Let $A$ and $\Phi $ be two independent random variables such that $E(A) = 0$, $E(A^2) < \infty$, and $\Phi$ is uniformly distributed ...
0
votes
1answer
60 views

What is a common way to measure the “goodness of fit” of an individual data point to a correlation?

Let's say I have a collection of data points (X & Y values) that show some correlation when, eg, Pearson's correlation formula is applied. What is a good measure for determining which data points ...
0
votes
1answer
342 views

MATLAB's implementation of cross correlation

Wikipedia gives the cross-correlation as $$ \begin{align*} (f \star g)[n] = \sum^{\infty}_{m = -\infty} f^{*}[m] g[n+m] \end{align*} $$ MATLAB's documentation gives ...