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

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Compute correlation between two random variables

A coin is flipped 100 times. Let $X$ be the number of heads in the first 70 flips and $Y$ the number of heads in the last 50. Compute the correlation of $X$ and $Y$. Here's my attempt: ...
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30 views

Correlation: How to extend it from pairs to further random variables?

How can one determine the correlation coefficients (or their intervals) between $n$ standard-normal random variables $X_i$, $i=1,...,n$, when $X_i$ correlates with $X_{i+1}$ with correlation ...
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65 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
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24 views

“correlation” between two subspaces

It is frequently important to determine the coherence of a matrix by finding the maximum pairwise correlation between all its column vectors. Similarly, when working with a union of subspaces it would ...
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100 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 ...
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2k views

How can I simply prove that the pearson correlation coefficient is between -1 and 1?

For building a recommendation system, I also use the Pearson correlation coefficient. This is the definition: $r(x, y)=\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^n ...
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45 views

calculate direct and indirect path coefficients

suppose we have following data wth mean and standard derivation corresponding ...
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46 views

Multiple regression and hypothesis test $H_0$:$\beta_2=0$

Multiple regression model $H_0$:$\beta_2=0$, $H_1$:$\beta_2 \neq 0$ where $\beta_2$ is the vector of elements ($\beta_2, \beta_3, \dots, \beta_k$) and $\beta$ is slope of regression line. Why it is ...
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46 views

Redundancies in covariance matrix

We know that covariance matrix is symmetrical. I have a vague intuition that there may be some other redundancies beyond that. For example, if A is correlated to B and B is correlated to C then A and ...
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1answer
657 views

Correlation coefficient of Wiener process

First, I'm not majoring mathematics. I'm studying economics and during reading a thesis I can't understand the 'wiener process' well. I read some books about it and understand the main idea and ...
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70 views

Finding Correlation between two types of Normally distributed data.

I have predictions and real outcomes for 6 projects of 2 different types (let's say type1 and type2). Is it possible to calculate the correlation between the two types? Predictions of the outcome: 3 ...
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687 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 ...
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64 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 ...
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60 views

A question about Pearson correlation coefficient

Suppose that we have two vectors $x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n)$ is the following correct about their Pearson correlation coefficient? $\operatorname{corr}(x,y)=\operatorname{corr}(x+a,y+b)$ ...
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1answer
30 views

Curve Fitting and Multiple Experiments

Say I do an an experiment 5 times, each of which gives you a list of data points. Do I fit a curve to each one separately and then average the parameters and their uncertainties? Or do I take the ...
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62 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 $n$, ...
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29 views

Finding correlation of data with potentially hidden time lags

Let's say I have few independent variables plus multiple observables that I monitor over time for a system. I'd like to find out if there is any correlation between the observables and any of the ...
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39 views

Multivariate Gaussian density from singular covariance

I have a multi-dimensional (~600dim) sample from which I determine its covariance matrix. The determinate of the covariance is 0. The sample does not show strong correlations when plotting 2 ...
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1answer
132 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 ...
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58 views

Expected value, correlation, and indepence.

I need help with a problem. Supposed x, y, and z are events in F (algebra of sets) in a probability space (universal set, F (algebra of sets), P). Define two random variables: a(omega) = ...
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1answer
69 views

Autocovariance of a given stochastic process?

I need to find the autocovariance $C_{YY}(t,s)$ of the stochastic process $Y(t) = t^2 X(t) -2X'(t)$ where $C_{XX}(t,s) = e^{-t^2 -s^2}$ is given. Using known properties I can calculate the ...
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201 views

Weighted least squares linear regression formulas using arbitrary weights

Say I have $n$ data points where each point has an $x$ value, a $y$ value, and an arbitrary weight value, $w$. The weight value is between $0$ and $1$, where $1$ means the data point will fully affect ...
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115 views

Why is this convolution true?

I am a little puzzled by how the following summation has been written as a convolution, with one of the inputs reversed in time. Consider the following sum on the LHS, and the convolution on the RHS. ...
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39 views

Question on change of variables during convolution/correlation

I am trying to understand how the following two statements are equivalent: $$ \sum_{l=-\infty}^{\infty} h^*[l] \ R_{xx}[m+l] = \sum_{i=-\infty}^{\infty} h^*[i-m] \ R_{xx}[i] $$ I get that we made ...
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42 views

A vector with fixed correlation with existing vector, is it always possible?

Suppose we have a known vector $X$ in $R^n$, and for any vector $Y$ in $R^n$, we impose on it the restriction that it must have a fixed correlation coefficient $r$ with $X$: \begin{align*} ...
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96 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 ...
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245 views

Proving $Y = aX + b$ given correlation coefficient $|\rho(X, Y)| = 1$

With correlation coefficient defined as: $$\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X)}\sqrt{\text{Var}(Y)}}$$ can you help me prove $$|\rho(X, Y)| = 1 \implies Y = aX + b$$
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141 views

Mean density of the nontrivial zeros of the Riemann zeta function

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ...
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1answer
373 views

Asymptotic correlation between sample mean and sample median

Suppose $X_1,X_2,\cdots$ are i.i.d. $N(\mu,1)$. Show that the asymptotic correlation between sample mean and sample median (after suitably centering and renormalization) is $\sqrt{\frac{2}{\pi}}$.
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94 views

Probability and correlation function, interpretation of a result

My question is originated from the paper ...
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1answer
88 views

Finding parameters for curve fitting

I have 500 observed data of variable $ x $ and corresponding $ y $. The functional model is where Is it possible to find suitable constants $ A , B $ ,$ \alpha , \beta $ so that the observed ...
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36 views

Kendall Tau distance over rankings containing different elements

Suppose to have two lists (or rankings) containing the same number of elements (but not the same elements). E.g.: $[5, 4, 3]$ vs. $[5, 4, 2]$ Dow do you define the Kendall tau distance between the ...
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3k 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 ...
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30 views

Correlation between properties

I have a table in a database where each record includes the ID of a person and the ID of a property of that person. Each person may have more than one property. Which is the statistical instrument ...
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1answer
184 views

Related rates, where do I start?

A revolving searchlight, which is $100$ m from the nearest point on a straight highway, casts a horizontal beam along a highway. The beam leaves the spotlight at an angle of $\frac{π}{16}$ rad and ...
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34 views

Finding correlations between many unknown functions.

Given an arbitrarily large number of black-box functions of one variable, is it possible to produce expressions that approximate their relationships to each other over their shared domain? Is it ...
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455 views

help understanding step in derivation of correlation coefficient

I'm looking to understand the starred step in the derivation below (also, if someone could help with the LaTex alignment, I'd appreciate it). The regression line is $y= b_0 + b_1 x$, where $b_0$ and ...
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84 views

Correlation function of Brownian motion. What am I doing wrong?

Can anyone tell me where I am going wrong here? (I am leaving out any random fluctuation forcings, because I don't think they are relevant to my problem.) 1: $\displaystyle \frac{dv(t)}{dt}=-\eta ...
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69 views

Calculation of conditional joint probability given certain conditionals for data which aren't independent

The context of this problem is the estimation of the distribution of a parameter $v$ given sets of data $A$ and $B$, where $A$ and $B$ are not independent. Suppose I know $P(v | A)$ and $P(v | B)$. ...
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91 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 ...
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140 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 ...
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57 views

Dimension free Concentration bounds for Martingales

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
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55 views

Finding the empirical correlation from a covariance matrix

I have this covariance matrix with five variables $X_1$ through $X_5$ in that order. ...
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177 views

Random walk serial correlation

Given a model $$Y_t =b_0 + b_1 \cdot X_t + b_2 \cdot Z_t + e_t,$$ where the error term $e_t$ follows a random walk form of serial correlation $e_t = e_{t-1} + u_t$. Further assume $u_t$ has zero mean ...
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265 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 ...
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137 views

Covariance and Correlation

Suppose there were m married couples, but d of these 2m people have died. Regard the d deaths as striking the 2m people at random. Let X be the number of surviving couples. Find: a) E(X) b) Var(X) ...
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194 views

Correlation of Indicator Variables

Show that for indicator random variables $I_A$ and $I_B$ of Events $A$ and $B$: $Corr(I_A, I_B) = Corr(I_A^c, I_B^c) = -Corr(I_A, I_B^c) = -Corr(I_A^c, I_B)$ Deduce that if $A$ and $B$ are ...
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422 views

Invariance of the correlation coefficient under linear transformations [closed]

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 ...
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1answer
61 views

How can I mathematically show the similarity between these 3 plots?

I have 3 3D plots of field strength measured around an antenna. I want to calculate the mathematical similarity between the points of the field patterns. How can I do this? thanks
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35 views

What values to choose for correlation?

To work out correlation I'm using the online calculator : http://easycalculation.com/statistics/correlation.php ...