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

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Problems about Farlie-Morgenstern family of bivariate CDFs

Hi I am trying to solve the following problem: Let $F_X:\mathbb{R}\to[0,1]$ and $F_Y:\mathbb{R}\to[0,1]$ be unnivariate Cumulative Distribution Functions (CDFs) and suppose $-1\le\alpha\le 1$. Define ...
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1answer
28 views

Correlation of three values

I have a data set that considers three values, $x,y,z$. And I have three questions: What's the relationship between $x$ and $y$? What's the relationship between $z$ and $y$? What's the relationship ...
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1answer
32 views

Why is autocorrelation used without normalization in signal processing field?

According to the wikipedia(Link), autocorrelation has two definition. Oh my god! In statistics, the definition of the autocorrelation between times $s$ and $t$ is like the following: $$\displaystyle ...
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1answer
32 views

Orthogonality of Two Signals.

My last question's link: Reconciling different definitions of orthogonality However, I failed to understand why they are equivalent. If $f$ and $g$ are real, \begin{align} ...
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0answers
26 views

How to calculate the correlation between two ratings

Suppose x and z are two persons who rate a classical piece of music y on a five point scale. x rates y $3$ out of $5$ z also rates y $3$ out of $5$ I need to know how much the ratings of x ...
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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 $$ ...
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1answer
18 views

Correlate normal shocks

I am trying to generate some random standard normal variables and correlate them In particular I want: $$ \bf Y \sim \mathcal N(0, \Sigma) $$ where $\textbf{Y} = (Y_1,\dots,Y_n)$ is the vector I ...
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0answers
39 views

Correlation between $X$ and $Y$ where $X$,$Y$ are from i.i.d. standard normals and $X+Y > 0$?

Suppose $Z = X + Y$, where $X$ and $Y$ are independent standard normal random variables. If we generate plenty of $(X,Y)$ pairs and only keep the ones where $Z>0$, what's the correlation between ...
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1answer
22 views

What is the correlation between the pairwise differences of 2 bivariate normal random variables? [closed]

Given (X,Y) bivariate normal, $U = \frac{X_i - X_j}{\sqrt2\sigma_x}$ and similarly $V = \frac{Y_i - Y_j}{\sqrt2\sigma_y}$ for any two independent pairs $(X_i, Y_i)$ and $(X_j, Y_j)$. Why is this true ...
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1answer
24 views

Sum of two standard variables with joint bivariate distribution?

Let $X_1$ and $X_2$ have standard normal distribution and let $(X_1,X_2)$ have a joint bivariate distribution. Can anyone explain why: $X_1+X_2=\sqrt{2+2\rho}Z$ where $Z\sim N(0,1)$ and $\rho$ is the ...
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2answers
32 views

Is there any difference between Correlation and Correlation coefficient?

I learnt in probability theorem class that correlation coefficient is $$ \rho=\frac{\sigma_{XY}}{\sigma_X \sigma_Y}=\frac{E\left[(X-\mu_X)(Y-\mu_Y)\right]}{\sigma_X \sigma_Y} $$ However, my ...
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2answers
29 views

How does Variance become an Autocorrelation Function?

"For a Gaussian stochastic process $X=\{X(t)|-\infty<t<\infty\}$ with mean function $\mu(t)=0$ for all $t$, its autocorrelation function is $$ E(X(t)\cdot X(s))=R(h)=\max(0,1-|h|), h=t-s. $$ ...
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1answer
35 views

Relation between correlation and regression

Let $y\in \mathbb{R}$ be a random variable. Let $y$ be expressed as a linear combination of $x_i$ $i=1,2,\cdots,n$, as follows \begin{equation} y = \sum\limits_{i=1}^nw_ix_i + \epsilon \end{equation} ...
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0answers
147 views

Can't find the relationship between two columns of numbers. Please Help [closed]

I cannot find the relationship between these two columns...other than I know that they both increase or decrease in value at the same time. I'm not a math person, but I would appreciate any help ...
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0answers
34 views

Autocorrelation function of a Wiener process & Poisson process.

Can anyone possibly explain step 3 and 4 in this solution?
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3answers
77 views

Correlation between $(X+Y)^2$ and $X$

Good evening everybody! I am totally puzzled about how to solve this problem. We are given the random variables $X$ and $Y$, both of which are independent and uniformly distributed on $[0,1]$. Then ...
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2answers
49 views

Autocorrelation function of integral of cont. white noise

Let $W(t)$ be continuous time white noise, that is, a wide-sense stationary (WSS) zero-mean Gaussian process with autocorrelation function $R_W (\tau) = σ^2\delta(\tau)$. Calculate the auto ...
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1answer
22 views

Need Help explaining this equation for the correlation coefficient

As I am not a math geek so I have problem comprehending this equation: equation for the correlation coefficient Its basically the formula used in the CORREL functon in Microsoft Excel and I am ...
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0answers
35 views

Why is the length of a BAPS (binary almost perfect sequence) a multiple of 4?

a BAPS is a binary sequence of length $n$ with an almost perfect autocorrelation: $$ A(\tau)=\left\{\begin{matrix} n & \tau=0 \\ k & \tau= {n \over 2} \\ 0 & else \end{matrix}\right. $$ ...
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0answers
15 views

Measuring correlation of two time series

I have two sets of annual time series (employment growth from 2 different sources), which I display by using index. I would like to measure somehow whether series 1 (which are more timely) can be used ...
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93 views

Minimum and maximum bound on mean of product of three pairwise uncorrelated random variables

There are three pairwise uncorrelated random variables $X, Y, Z$ $$E(X) = E(Y) = E(Z) = 0$$ $$E(X^2) = E(Y^2) = E(Z^2) = \sigma^2$$ How we could find minimum and maximum bound on $E(XYZ)$? I ...
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0answers
19 views

Show covariance matrix must have condition number larger than correlation matrix

The inverse of covariance matrix can be used to find the conditional independencies among variables. This inverse is more sensitive to changes then the correlation matrix. As a result two samples with ...
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0answers
20 views

General correlation between function with itself and other input data

I've collected data for a function F = f(g(x)), for different function shapes g(x). The goal is to predict values of ...
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1answer
32 views

Correlation Coefficient

I am trying to understand the following equation for Correlation Coefficient: $r = \frac{\sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)}{\sqrt(\sum_{i=1}^{n}(x_i - \bar x)^2\sum_{i=1}^{n}(y_i-y)^2)}$ ...
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1answer
23 views

calculate mean of two variables given two regression equations

Given these 2 regression equations how do I compute mean and find $r_{XY}$. $X=-0.4Y+6.4$ $Y=-0.6X+4.6$ when I rearranged the equations, I solved for $X$ and $Y$ hence $X=6$, $Y=1$ How do I get ...
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0answers
85 views

Minimal conditions to show $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below $$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$ $$G = ...
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1answer
40 views

Constructing a sample by correlation

Suppose we have two samples with known correlation (should be relatively high). Say both samples have $n$ data points. What if now we still know the correlation factor but one sample only consistent ...
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1answer
25 views

wss process and autocorrelation

In a paper, I saw a quiz question about auto-correlation of a WSS process than I can not understand. It says: Let $X(t)$ be WSS. Which of the following can be correct? a) $E[X(t_1) X(t_2)] = |t_1 - ...
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2answers
71 views

Uniform Random Variable: Correlation and Independence

Let X be a uniform random variable defined on the interval $(0,1)$. If $Y = 6X^2−6X+1$, compute the correlation of X and Y . Are X and Y independent? Are X and Y uncorrelated? So my work is. $F(X) ...
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1answer
71 views

Positive definiteness of a correlation matrix

With $n$ natural numbers 1, 2, ..., $n$, there are $N=n(n-1)/2$ unique pairs that are the 2-D indices for random variables $z_{ij}$ ($1 \le i < j \le n$). I have the correlation matrix $A$ of ...
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0answers
14 views

Mean value given error is distributed ~ N( 0 , .1 )

The true weight of an object is w It is weighed two different times X1 and X2. Then X1 = w + E1 and X2 = w + E2 Where E are the two measurement errors. Suppose the error is iid with a mean of 0 ...
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0answers
29 views

Gaussian Copula Function

I have a question that I believe I know the answer to, but I would like to double check. Is it possible to use a copula function to calculate the joint probability of three or more variables? I'm ...
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0answers
20 views

Correlation coefficient of a block matrix

My question is related to the computation of correlation coefficient of a block covariance matrix. The correlation coefficient can be computed as: r = $ cov(X,Y)/std(X) std(Y) $ But if I have a ...
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0answers
34 views

Probability of at least one event ocurring given correlations

I have a problem that I can't solve. I have three variables. Let's call them $A$, $B$, and $C$, and the specific probabilities of them ocurring, $\text{P}(A)$, $\text{P}(B)$, $\text{P}(C)$. I also ...
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0answers
17 views

How to derive distribution of correlation coefficients under bivariate normal condition

X and Y are bivariate normal and r is the correlation coefficients. If X,Y independent. I want to show : $\frac{r}{\sqrt{(1-r^2)/(n-2)}}$ follows $t_{n-2}$ distribution.
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19 views

Proof about the ranks of a correlation matrix

$C$ is the correlation-matrix of the random variables $B_1,B_2,\dots,B_n$. These random values are the columns of matrix A. I calculate the elements of $C$ using this formula: Where I have to ...
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0answers
39 views

How to Discover Function Relating Two Variables

I have two variables, call them $x$ and $y$, that can possess any arbitrary positive integer value. I have an automated process that produces an output based on these variables. I do not, however, ...
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0answers
14 views

Point Biserial Correlation

I had a look around here and around the web and some literature but couldn't find any questions on this topic. I would like to quantify the strength of association between the the 2 variables X and ...
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1answer
25 views

Correlation of two random variables

A random sample of $100$ variables is given. Each of them is independent and identically distributed with $N(0,1)$. What is the correlation between sum of $98$ variables and sum of $100$ variables?
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1answer
43 views

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}_i) = 0$

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}) = 0$ in the image below (third and fourth line of the proof!). Why?
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0answers
18 views

Min/Max variance of 3 correlated variables

You have 3 different random variables (assets) all with exactly the same variance. What is the maximum and minimum variance of the 3 variables (assets) combined? Proposed ...
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0answers
28 views

testing correlation coefficient in a bivariate normal distribution

How can I show that $\dfrac{\hat{\rho } \sqrt{N-2}}{\sqrt{1-\hat{\rho}^2}}$ has a t-student distribution with $N-2$ degrees of freedom. I think I have to write it as a quotient of a normal $(0,1)$ ...
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1answer
27 views

Random Variables, Variance and Coefficients

So, I'm trying to get my head around a question and I was wondering if you could help me. Given that the correlation coefficient $\rho$ for random variables $X$ and $Y$ is: ...
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19 views

explanation of correlation of stationary stochastic processes

I have some questions about correlation in stationary stochastic processes. I know that the expectation of a random variable is $E(x)=\int_{-\infty}^{+\infty} a ...
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0answers
24 views

Proof that correlation coefficient squared equals the coefficient of determination

Hi I as the title says I'm looking at the proof that $r^2$ = $R^2$ in the case of simple linear regression, but I don't understand one part. There are different versions of the proof, but in most of ...
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1answer
39 views

Probability at least one, using correlation

I have a problem using the correlation in combination with the "at least one" probability. I have $P(A)=57\%$, and $P(B)=74\%$, and I calculated their correlation coefficient and it is $0.1557$. To ...
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1answer
19 views

How to find the percent variation in Y is explained by X?

I know that the r^2 value for the data is 0.9832. Is there a way to use that value to find the percent variation in Y is explained by X? Or do I need to use the data given to me?
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2answers
31 views

Uncorrelated and X given $Y = 0$

Is the following true or false? Suppose that $X$ and $Y$ are two discrete random variables defined on the same probability space. If $E[X] = E[Y] = 0$ and $E[X | Y=y] = 0$ for all $y\in Y$, then $X$ ...
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1answer
66 views

Decorrelating variables using Cholesky decomposition

I am looking for a method to decorrelate several variables, so that their covariance matrix is diagonal, while keeping the original mean for each of them. I found this old article which seemed pretty ...
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1answer
130 views

Finding the autocorrelation of a sine wave.

The autocorrelation of sin(t) is defined as $$\displaystyle \int_{-\infty}^{\infty} \sin(t+\tau)\sin(t)d\tau$$ I've tried using the Wiener-Khinchin theorem which says that ...