1
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0answers
15 views

Bounding the Correlation Coefficient

Let us assume we have two random variables $X$ and $Y$ where $X = f(A, B, C)$ and $Y = g(A, B, C)$. $A, B, C$ are 3 independent random variables and the functions $f, g$ are known but rather expensive ...
0
votes
0answers
7 views

Methods for Uncorrelating data - Comparison

I see that both PCA and Cholesky Decomposition could be used for uncorrelating correlated data. When should one be used? What are the assumptions made by each model. When do the methods fail? Are ...
0
votes
0answers
16 views

Generate Correlated Normal and Log-Normal Random Variable

The standard approach for generating two normally distributed random variables some with correlation $\rho$ is explained here: Generate Correlated Normal Random Variables. Now let $X,Y$ be normally ...
0
votes
0answers
9 views

Difference in magnitude between two cross-correlations by two different way of calculations.

I think there are two ways of calculating cross-correlations for two difference random variables, X and Y. I am assuming discrete functions. 1) Multiplication $$ \sum_{m=-\infty}^\infty x[m]y[m+n] ...
0
votes
1answer
58 views

Summing dependent random variables with unknown joint cdf

Suppose that X_1, X_2,... X_5000 are discrete and dependent non-identically distributed random variables, whose marginal distributions are known, but whose joint distribution is not known. Is there ...
1
vote
0answers
12 views

Maximum correlation of n variables

For $n>2$ variables, one cannot arbitrarily choose the correlations $\rho_{ij}$ because the resultant correlations must obey the law of cosines. Equivalently, the covariance matrix between them ...
2
votes
1answer
40 views

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$ is a convex risk measure, but it fails the subadditivity property in order to be called coherent. A mapping ...
1
vote
4answers
440 views

Inferring covariance cov[X,Z] from cov[X,Y] and cov[Y,Z] of known distributions

Suppose X, Y and Z are real random variables of known distributions. If one knows the covariance $COV(X,Y)$ and $COV(Y,Z)$, is it possible to infer $COV(X,Z)$?
0
votes
1answer
45 views

How to deal with the following problem of correlated random variables?

I have the following information: $\left[ \begin{array}{l} {X_1}\\ \vdots \\ {X_K} \end{array} \right]$ are correlated random variables with (zero mean, unit variance) covariance matrix $\left( ...
2
votes
2answers
96 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 ...
0
votes
2answers
1k views

Expected value of two dependent variables is still a product of expectations

For independent variables we have $E[XY]=E[X]E[Y]$. Now, since I could not find a statement that the converse is also true, I suspect that there are examples of dependent variables where this relation ...
0
votes
1answer
23 views

Correlation formula for discrete phenomena in time

I need a statistical formula to capture a particular phenomena that I need to model in software. I have a light that can be on or off. When turned on, it can be one of many colors (for example, ...
0
votes
1answer
107 views

Random Variable Problem w/ variance

Three zero mean, unit variance random variables X, Y, and Z are added to form a new random variable, W = X + Y + Z. Random variables X and Y are uncorrelated, X and Z have a correlation coefficient of ...
1
vote
2answers
766 views

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 ...
2
votes
0answers
65 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 ...
2
votes
0answers
68 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 $n$, ...
1
vote
1answer
153 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 ...
1
vote
1answer
106 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 ...
5
votes
1answer
4k views

Generate Correlated Normal Random Variables

This will be a difficult question to explain, but I'll give it my best. I'm running a simulation with a group of objects (let's just call them agents) and each agent has $n$ parameters that defines ...
1
vote
1answer
92 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 ...
0
votes
2answers
149 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 ...
1
vote
1answer
311 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 ...
1
vote
1answer
125 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 ...
0
votes
1answer
81 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 ...
0
votes
1answer
108 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 ...
1
vote
0answers
22 views

Estimating the likelihood of independence of two discrete variables using the co-occurrence count matrix.

I have some data about users from different regions visiting different directories of some website. Aggregating that data I get the co-occurrence frequency matrix (for regions and directories). Now I ...
3
votes
4answers
5k 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 ...
3
votes
1answer
132 views

Autocorrelation of wrapped Wiener process

Let $\phi(t)$ be a Brownian Walk (Wiener Process), where $\phi\in[0,2\pi)$. As such we work with the variable $z(t)=e^{i\phi(t)}$. I would like to calculate $$E(z(t)z(t+\tau)).$$ This is equal to ...
1
vote
1answer
52 views

How can I show that $z_i =\cos(iw)$ where $w$ is uniform on $[0,2\pi]$ is a white noise process?

How can I show that $z_i =\cos(iw)$, where $w$ is uniform on $[0,2\pi]$ is a white noise process? So far, I have shown $E(z_i)=0$ by integrating. However, I need to show ...