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

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3
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3answers
501 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 ...
0
votes
1answer
421 views

Invariance of the correlation coefficient under linear transformations [on hold]

Show that for arbitrary random variables $X$ and $Y$, and constants $a,b,c,d$ with $a$ and $c$ nonzero, $$ \mathrm{Corr}(aX+b,cY+d) = \begin{cases} \mathrm{Corr}(X,Y)\quad&\text{if }a \text{ and ...
4
votes
3answers
741 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 ...
1
vote
2answers
720 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!
5
votes
1answer
144 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, ...
3
votes
0answers
134 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
255 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 ...
2
votes
1answer
159 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 ...
2
votes
2answers
7k 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" ...
1
vote
1answer
122 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
1answer
95 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
0answers
115 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
400 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
153 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
104 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
56 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 ...