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

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2
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1answer
137 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 ...
3
votes
3answers
101 views

Correlation of uniform variables

Let $X$ and $Y$ be independent random variables, $X,Y \sim unif(0,1)$. Let $U = \min \{X,Y\}$ and $V = \max\{X,Y\}$. Find the correlation coefficient of $U$ and $V$. I think we can assume that $U = X$...
1
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0answers
49 views

How can I get a Covariance Matrix from Mean and Variance?

this may be a very basic question. I have the mean and variance for 12 lognormal distributions: ...
0
votes
1answer
44 views

Relation between Regularization and correlation

I was going through Chapter 3 (page 63 bottom) of Elements of Statistical Learning. While explaining regularization in ridge regression authors make the following statements. "When there are many ...
1
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0answers
46 views

centering two variables $X$ and $Z$ makes $cov (X,XZ) = 0$

I've read that centering two normal (or symmetrical) variables $X$ and $Z$ affects correlation of centered $X$ with interaction term $X\cdot Z$ in such way, that this correlation $cor(X-EX, X\cdot Z)$ ...
0
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0answers
297 views

How to describe the correlation between two non-random variable mathematically?

As we all know, correlation is a statistical relationship between two random variables. However, if there are two non-random variables, is there correlation between them, if it has, how to describe it ...
0
votes
2answers
62 views

Correlation between two variables

Assume $X_1$, $X_2$, $X_3$,..., $X_n$ are i.i.d, say that $Y_1$ = $X_1^2/\sum_i X_i^2$ and $Y_2$ = $X_2^2/\sum_i X_i^2$, how to calculate the correlation between $Y_1$ and $Y_2$ or prove that they are ...
1
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1answer
20 views

Correlated explanatory variables in linear regression

Is it any reason to assume that if two strongly correlated explanatory variables have impact on response that regression coefficients for these variables have the same signs ? Could such assumption be ...
0
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1answer
43 views

Interpreting high p value and low correlation value

I am trying to run regression on financial data in R. I am new to regression analysis so I am finding it to difficult to interpret certain scenarios. I have the code as follows: Regression analysis ...
1
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1answer
33 views

Formula for the correlation between two different variables

Jon planted a plant. When the plant grew to $4\,cm$ tall he decided to start to measure how much the plant grew each week. Here are Jon's measurements: Week $0$: $\;\;4\,cm$ Week $1$: $\;\;...
0
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0answers
139 views

Comparing two vectors based on order and ranking?

What I want to do is compare the ordering of variables determined by the ranking of each variable. For example: Say, I have a rating system that is made up of 5 different ratings - Excellent Good ...
0
votes
1answer
479 views

Relation between Correlation and Convolution

We have two functions of time $f(t)$ and $g(t)$, for which convolution and correlation are defined as following: Convolution: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$ ...
0
votes
2answers
461 views

Why is the maximum value of cross-correlation achieved at similar section?

I'm a bit confused and probably need some sleep. When trying to find a short signal inside a long one (or the delay), it's almost a trivial fact that we should look for the maximal valued coefficient ...
1
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0answers
35 views

Multiple variables correlation

There are a number of entities. Each entity has three sets of parameters. Each set of parameters describes entity behaviour in a specific system. The problem is to find mathematical method that would ...
1
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0answers
78 views

Partial Correlation Coefficient

I have the following questions on computing the correlation coefficient. Let us say we have two discrete random variables $X_1$ and $X_2$, where $X_1$ has $n_1$ outcomes and $X_2$ has $n_2$ outcomes....
0
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1answer
66 views

We have an urn with $5$ blue balls and $15$ red balls.

We remove $7$ without replacement. Let $R$ be the number of red balls removed and $B$ the number of blue balls removed. Do you expect $R$ and $B$ to be positively correlated, negatively correlated, or ...
0
votes
1answer
80 views

Given an unfilled pmf, How to compute the Correlation coefficient?

This is the format in which I was given the PMF. Given this pmf $$\begin{array}{lll} x&y&f_{xy}(x,y)\\ \hline 1&1&.25\\ 1&2&.25\\ 2&1&.25\\ 0&0&.25 \end{array}...
0
votes
1answer
51 views

Can I sum variances to a compound variance?

Say I have three locations A,B,C and I have a person going from A to B and measure the time it takes. Same for B to C. Let the variance of the time it takes for the path AB be a and for BC b. Is it ...
0
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1answer
63 views

Approximate as Independent Identically distributed

If $N$ random variables are identically distributed but weakly correlated, in what condition we can approximate them as independent identically distributed (iid) ? I saw an old paper where based on ...
0
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1answer
302 views

The autocorrelation function - the result in the form of a vector.

I've implemented the autocorrelation function in Python according to the normalized autocovariance function for discrete signals, i.e: $$\gamma(k)=\frac{1}{N-1}\sum_{i=0}^{N-k}(x(i+k)-x_{s})(x(i)-x_{s}...
-1
votes
2answers
630 views

The product of a normal and Bernoulli variables, independent from each other

Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that: \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} & \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 & \mbox{...
1
vote
1answer
50 views

If two functions are even, then X and Y are uncorrelated

I Need some help: Let $Y=h(X)$ be a real square integrable function and X has a density function $f$. Show: If $f$ and $h$ are even functions then $X$ and $Y$ are uncorrelated (but generally not ...
0
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0answers
58 views

correlation coefficient is over 0.7

"correlation coefficient can be over 0.7 then we can say Two factors have some strong relation. Then What is the 'reason' that we can say like that? Explain it." I got that problem and I really dont ...
0
votes
1answer
132 views

Proof of Correlation Coefficients

Good evening, I have a problem with an exercise: Let $X$ and $Y$ be two real square integrable random variables with var$X>0$, var$Y>0$. The correlation Corr$(X,Y)$ quantifices how far $X$ and ...
0
votes
1answer
42 views

Inverse Z transform of symmetric function $R_{x}(n) = 3\cdot (0.8)^{|n|}$

On Z-transform table, most of the pairs are only valid for $n≥0$. My question is to find PSD (Z transform) of $$R_{x}(n) = (0.8)^{|n|}$$ Note that $n$ is an integer span from $-\infty$ to $\infty$....
1
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1answer
59 views

Three pairwise uncorrelated random variables

Given $\xi$, $\eta$, $\zeta$ are pairwise uncorrelated, can we say, that $E(\xi\eta\zeta) = E\xi E\eta E\zeta$?
0
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0answers
30 views

Why does one compute the power spectrum of an image from the Fourier transform of its autocorrelation and from the square of its spectrum?

image: f(x,y) fourier transform of f is F(u,v) my Goal is to compute its power spectrum. [denoted by P(u,v)] the first way to compute is by using the magnitude of fourier transform: $|F(u,v)|=\sqrt{...
0
votes
1answer
24 views

Help with correlation question? How to solve this?

Let $X$ and $Y$ be random variables and $a,b$ $\in$ $\mathbb{R}$ such that $a \neq 0$. If $Y = aX + c$, then show that corr($X, Y$) = +1 or corr($X, Y$) = -1.
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0answers
25 views

Weighted Average of Correlation Matrix

Let $R$ and $Q$ be two correlation matrices of the same size and let $p\in[0,1]$. I'm trying to show that $pR+(1-p)Q$ is still a correlation matrix. I claim that $\sqrt pX+\sqrt{1-p}Y$ is a vector ...
0
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1answer
33 views

Joint PDF Correlation

In the problem I am given $f(x,y)=2,\ 0 < x < y,\ 0 < y <1$. I'm trying to find the correlation $\rho$ which I know is equal to $$\rho = \frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}$$ ...
1
vote
1answer
63 views

Computing the expected value of the product of two discrete variables

I didn't know why I compute $E(XY)$ wrongly. $$X=(1, 2, 0.5, -1),\qquad Y=(-2, 1, -0.5, 2).$$ $$E(XY) = \frac{-2 + 2 -0.25 -2}{4} = -0.5625\text{ (incorrect)}$$ because $\operatorname{Cov}(X,Y)=-0....
1
vote
1answer
44 views

Size of sample and correlation coefficient

$X$ and $Y$'s correlation coefficient is $r=0.5$. What is the size of sample when the correlation is significant at $\alpha=0.05$ with two sided test? Is there a more "formal" way to solve this ...
0
votes
2answers
236 views

Partial proof for correlation coefficient formula?

I've been working to prove the formula for the correlation coefficient, since asking my last question yesterday (Meaning of denominator in correlation?). If this post in any way violates any ...
1
vote
2answers
440 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, ...
0
votes
1answer
104 views

Correlation coefficient and Expectation of two dimensional normal distribution.

Random variable (X,Y) is normally distributed. Conditional expectations are $E(X|Y=y)=0.25y + 2$ $E(Y|X=x)=x-2$ How can i determine correlation coefficient and when that is known, the expectations ...
0
votes
1answer
88 views

rewriting formula containing covariance and variances

just trying to follow a formula. the equation starts of as follows, 1 = sum( xi * (cov(ri, r) / sigma^2(r) ) please note i's are subscripts then next line ...
1
vote
1answer
63 views

What is the expectation of the product of dependent, normal random variables

Question: Let's say I have $X \sim N(\mu_1, \sigma) $ and $Y \sim N(\mu_2, \sigma) $. I know that $ cor(X,Y) = \rho $. What is $E(XY)$? What I've tried Based on a similar question where X and Y are ...
1
vote
2answers
183 views

relationship between multiplication and correlation

is there a deep interpretation of multiplication as correlation? is this in some sense the most fundamental way to "combine" objects (eg numbers) into products? my reasons for asking are that the ...
0
votes
1answer
49 views

Correlation between 3d images and their slices

I work in the field of the image processing and I need to compare results of my algorithm with a gold standart results. For this purpose I calculate the Pearson correlation coefficient between the ...
1
vote
2answers
46 views

Show $E\left(\mathbf{X}_i \otimes \mathbf{u}_i\right)=\mathbf{0}$ implies $E\left(\mathbf{X}_i^{\top}\mathbf{G}\mathbf{u}_i\right)=\mathbf{0}$

Let $\mathbf{X}_i$ be a $G \times K$ random matrix, and let $\mathbf{u}_i$ be a $G \times 1$ random vector, and suppose we have a sample of $i=1,\ldots,N$ of each. Suppose the following condition ...
1
vote
0answers
140 views

Definition of Autocorrelation Function (ACF)

For a weakly stationary time series $\left\{r_{t}\right\}$, the definition of ACF is (from Ruey Tsay's "Analysis of Financial Time Series") $$ \rho_l=\frac{Cov(r_t,r_{t-l})}{\sqrt{Var(r_t)Var(r_{t-l})}...
0
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0answers
198 views

Finding a relative error measure on a data set proportional to another

I have a set of exact data points $\mathcal{X}=\{X_i\}$ and another approximate one $\mathcal{Y}=\{Y_i\}$ where there is a correspondence between $X_i$ and $Y_i$ for all $i$. If $\mathcal{Y}$ was ...
1
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0answers
13 views

Proving $Corr(\hat{e}_{ij}, \hat{e}_{jk}) = \frac{-1}{n_i-1}$ for $ j \neq k$

For the model of a single factor experiment: $y_{ij}= \mu + \alpha_i + e_{ij}$, $(1 \leq i \leq a, 1 \leq j \leq n_i)$, where a = the number of treatments, $n_i$ = the number of experimental units ...
1
vote
2answers
163 views

The inverse of AR structure correlation matrix / Kac-Murdock-Szeg ̈o matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots &\...
0
votes
1answer
553 views

What does the multiplication of standard deviation of two variables gives?

If we need to find the correlation between two variables it is given by the formula - co variance of two variables divided by the multiplication of Standard deviation of the two variables. My ...
2
votes
1answer
38 views

Question on the correlation between two dependant variables

I'm working on this question and it's stumping me. Let Sn = X1 + ... + Xn (with n>=1) be a random walk with X1,...,Xn be iid RV's. E(Xk)=mu Var(Xk)=sigma^2. Find the covariance of Sn and Sm Can ...
-2
votes
1answer
74 views

Combination problem: random selection in a group

A scientific committee of 4 persons is to be randomly selected from a group consisting of 3 biologists, 3 physicists and 4 mathematicians. Let X denote the number of biologists, Y the number of ...
2
votes
0answers
37 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = \...
2
votes
0answers
47 views

OLS standard error that corrects for autocorrelation but not heteroskedasticity

Question: By mapping the OLS regression into the GMM framework, write the formula for the standard error of the OLS regression coefficients that corrects for autocorrelation but not heteroskedasticity....
2
votes
1answer
35 views

Is $d(i,j) = 1-\textrm{corr}(i,j)$ a metric?

I need to make sure that this function is a metric: $d(i,j) = 1-\textrm{corr}(i,j)$ where $\textrm{corr}(x,y)$ is the Pearson correlation coefficient which ranges from $[-1,1]$. With this scaling I ...