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

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115 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 ...
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1answer
266 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: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$ ...
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2answers
291 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 ...
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0answers
33 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 ...
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71 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 $n_2$ ...
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1answer
64 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 ...
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1answer
70 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 ...
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1answer
46 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 ...
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1answer
59 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 ...
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1answer
246 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: ...
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2answers
365 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 & ...
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1answer
47 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 ...
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0answers
40 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 ...
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1answer
112 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 ...
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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 ...
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1answer
50 views

Three pairwise uncorrelated random variables

Given $\xi$, $\eta$, $\zeta$ are pairwise uncorrelated, can we say, that $E(\xi\eta\zeta) = E\xi E\eta E\zeta$?
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26 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: ...
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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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24 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 ...
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1answer
29 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 $$\rho = \frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}$$ ...
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1answer
50 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 ...
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1answer
43 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 ...
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2answers
214 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 ...
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2answers
335 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, ...
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1answer
78 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 ...
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1answer
74 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 ...
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1answer
52 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 ...
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2answers
130 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 ...
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1answer
42 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 ...
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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 ...
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101 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") $$ ...
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164 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 ...
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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 ...
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2answers
104 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 ...
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1answer
505 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 ...
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1answer
35 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 ...
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1answer
71 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
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0answers
36 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) = ...
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0answers
40 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 ...
2
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1answer
33 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 ...
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0answers
35 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 ...
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0answers
155 views

Difference between identity and diagonal covariance matrices

thanks in advance for the help. Suppose I am training a linear model. What are the conceptual differences between using a diagonal covariance matrix and the identity? It is clear to me that the ...
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1answer
24 views

Covariance matrix computed based on a covariance function

I am reading Chapter 4 of Gaussian Processes for Machine Learning. It says that a matrix $K$ whose entries are computed as $k_{ij} = k(x_i, x_j)$ where $k$ is a covariance function is a positive ...
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1answer
115 views

Constraints on correlation coefficients of multiple random variables

I am looking for a generalization of Correlation between three variables question for more than three variables. It is stated in one of the answers there that, for three variables with pairwise ...
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2answers
299 views

How to find relation between 2 numbers

I have been practicing programming for many months now and what I found difficult is not about solving problem. But it is how to find the "how to solve problem" to make computer solves that for me! ...
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2answers
49 views

Correlation of random variables with joint PDF proportional to $x^{a-1}y^{b-1}(1-x-y)^{c-1} $

The random variables $X$ and $Y$ have joint PDF $$f(x,y)= \frac{\Gamma(a+b+c)}{\Gamma(a)\Gamma(b)\Gamma(c)}x^{a-1}y^{b-1}(1-x-y)^{c-1} $$ where $0 \leq x \leq 1 , 0 \leq y \leq 1, x+y < 1 $ where ...
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0answers
151 views

Checking hand manipulations of matrices

Beginning with a 4*3 matrix: 5 4 -1 2 3 -3 3 4 -4 1 3 -2 I have to perform four manipulations on it, which I did by hand. I wanted to ask if my thinking and/or ...
2
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2answers
136 views

What is the intuitive meaning of uncorrelated?

I was going through the derivation of the Kalman filter and it mentions that since noise (v) is uncorrelated to the state (x) and the state estimate (xbar), the following quantity is zero: E((x - ...
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2answers
60 views

Probability of observing a false correlation and confidence limits

In oil and gas exploration/development it is common to use acustic impedance derived from reflection seismic surveys to predict the porosity measured in wells drilled in the reservoir. I often use ...
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1answer
20 views

Re-calculating Value of $100 in Each State by Specific State

I'm using this Tax Foundations graphic for data. How would I re-calculate each state based on a specific state? For example, what if I wanted to base the control state on Missouri, which is $113.51. ...