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

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Covariance of $10$ Coin Flips

I'm getting the hang of using the properties of Covariance to make calculating it much easier but I'm stuck on this one. Fair coin tossed $10$ times. Let $X$ denote number of heads observed and ...
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2answers
113 views

Covariance of three dice rolls

I understand this question has been asked but I have a different comment to make on the matter and wondering if someone could help me. Let Z1,Z2,Z3 be values resulting from three tosses. ...
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1answer
38 views

Simple linear regression prove variables are uncorrelated:

I am working on the following problem: In a problem of simple linear regression, $$Y = \hat\beta_0 + \hat\beta_1 x(bar),$$ show that the random variables $\hat\beta_1$ and $Y$ are un-correlated (All ...
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1answer
98 views

Fair Coin Covariance

Consider an experiment in which three fair dice are tossed simultaneously and independently. Let $Z_1,Z_2,Z_3$ be the values resulting from the three tosses. Define $X=Z_{21}+Z_{22}−Z_{33}$ and ...
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1answer
40 views

What does the “empirical” autocovariance function represent?

My professor gave me the sequence ${X_n} = {1,5,5,1,5,5,...}$ and asked us to compute the empirical autocovariance function given below. $$\displaystyle \hat \rho(1) = \lim_{N \to \infty} \frac{1}{N} ...
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2answers
43 views

Is the autocovariance function of a sequence identically zero if the sequence is iid?

My professor gives the following definition for the autocovariance function. $$\rho(i,j) = Cov(X_i , X_j)$$$\\$If I have a sequence that is iid, when i compute $\rho(n,n+1)$ for $n \geq 0$, I found ...
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1answer
810 views

Distribution of the sum of normal random variables

Let $X\sim \mathcal N(\mu_X,\sigma_X^2),\ Y\sim \mathcal N(\mu_Y,\sigma_Y^2)$ two normal random variables and $a,b\in \mathbb R$. If $X,Y$ are independent, then $$aX+bY\sim \mathcal ...
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0answers
23 views

Detecting camera shake

I have a bunch of data captured from a worm tracker that consists of a B&W camera that stares down at a few dozen worms for an hour at a time. The tracker captures the outline of each worm ...
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1answer
308 views

Autocorrelation and spectral density in MATLAB

This question is threefold. We have an LTI system that is a first degree Butterworth LP filter with the power TF where fu = 110Hz and ...
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1answer
757 views

Autocorrelation and spectral density in MATLAB

This question is twofold. We have an LTI system that is a first degree Butterworth LP filter with the power TF where fu = 110Hz and ...
0
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1answer
26 views

Autocorrelation of a sequence of vectors

Let's say I have a sequence of 2-d vectors and I want to calculate autocorrelation of this sequence of vectors. If $V_i$ where i = 1:n is the list of vectors then acf as a function of time lag 't' is ...
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1answer
42 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 ...
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0answers
185 views

Generate correlated random numbers precisely

Let's assume I want to generate k samples of n random numbers, that are correlated according to a given correlation matrix C (e.g. $n = 3$): ...
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1answer
41 views

Correlation: Concept to Formula

In digital signal processing, we calculate the correlation between two discrete signals by multiplying corresponding samples of the two signals and then adding the products. Where does this ...
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1answer
67 views

probability need help on correlation problem [duplicate]

A deck of 52 cards is shuffled you are dealt 13 cards. Let $X$ and $Y$ denote, respectively, the number of aces and the number of spades in your hand. Show that $X$ and $Y$ are uncorrelated. I try to ...
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1answer
57 views

finding the unspecified ${\bf E}[X]$ and $\rm var(X)$ given the expectation of higher powers of $X$

Homework Problem: It is known that a for a standard normal random variable $X$, we have ${\bf E}[X^3]=0$, ${\bf E}[X^4]=3$, ${\bf E}[X^5]=0$, ${\bf E}[X^6]=15$. Find the correlation coefficient ...
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4answers
826 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 $COV(X,Z)$?
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1answer
1k views

How to construct a covariance matrix from a 2x2 data set

so if given a covariance matrix I can find the eigenvalues and move forward from there... but I seem to have trouble with the step before if I am given a data set and am told to create the covariance ...
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1answer
61 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( ...
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0answers
72 views

Show that a function is log supermodular

I have been struggling with the following Let $X$ be finite and a poset $P = (X, \leq)$, and for any $A \subseteq X$ we can define the function $f_A$ on $\mathcal{P}(A)$ as follows $$ f_A(Y) = \#\{ ...
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1answer
564 views

Correlation between complex random variables

I am struggling to find the correlation between two complex r.vs; X and 1/Y i.e. E{X*/Y}, where '*' denotes the conjugation operator. The complex r.s X and Y are correlated with each other with known ...
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2answers
99 views

Correlation coefficient

I'm a little puzzled by the whole random variable thing. I've got two random variables, $\mathcal{X}$ and $\mathcal{N}$, both with gaussian distribution with mean = 0 and $\sigma_{\mathcal{X}}^2$ and ...
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1answer
58 views

Estimate correlation coefficient of unknown variable

Consider variable y depends on variable x and z linearly. I have $100$ sample values of $y$ and corresponding $x$ but don't have any values of $z$. The functional model is $$y = \alpha_1x + \alpha_2z ...
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3answers
29 views

Correlation and what it tells me

OK, I need a little help here. I have attached two pictures; Data and Chart in which the data shows a correlation coefficient of 0.283168 which was calculated by Excel. Can someone please tell me ...
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1answer
232 views

Correlation Coefficient Distribution Function: An Apparent Discrepancy?

I'd like to explain an apparent discrepancy between: (1) The sample correlation distribution function between sample vectors for a bivariate, correlated random variable (correlation coefficient = ...
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0answers
88 views

Spatio-temporal triple correlation

I would like to simplify if possible the spatio-temporal triple correlation of the following function: $$f(\vec{x},t)=\delta(\vec{x}-\vec{x}_0(t)) \otimes f_p(\vec{x})$$ where $\delta$ is the Dirac ...
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0answers
32 views

What is correlated with what in a linear regression?

I'm trying to make sure I understand the ins and outs of a linear regression and am making a table for what is correlated with what, so just want to see if I have everything included. I'm looking at ...
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1answer
81 views

How can we derive expectation of two dependent normal distribution?

$\mathbf{X}$ and $\mathbf{Y}$ are each dependent normal random variable, then how can we derive like this one? $$\mathbf{E}\{e^{\mathbf{X}}e^{\mathbf{Y}}\}$$ I know the each first moment is ...
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0answers
105 views

Using mutual information to estimate correlation between a continuous variable and a categorical variable

As for the title, the idea is to use mutual information, here and after MI, to estimate "correlation" (defined as "how much I know about A when I know B") between a continuous variable and a ...
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0answers
34 views

Autocorrelation Clarification

Could anyone help clarify a high level explanation of autocorrelation? I understand that it is a measure of correlation between a timeseries and a lagged version of the same series. If we have take ...
2
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1answer
954 views

Maximum and minimum Correlation Coefficient

I have a question regarding the correlation coefficient. The inspiration is from a story where a student collected a set of $(X,Y)$ pairs, but lost the pairings. Hence, he is left with two sets of ...
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2answers
154 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 ...
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1answer
91 views

Generating correlated random numbers from Normal Distributions

If I have a sequence taken from X~N (μ1 , σ1 ). Is it possible to generate a sequence of numbers drawn from Y~N (μ2 , σ2) such that X and Y have correlation ρ?
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2answers
4k 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 ...
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1answer
153 views

Countermonotonicity and minimum linear correlation coefficient

In an example exercise they question whether it is possible to construct a bivariate distribution of $LN(0,1)$- and $LN(0,4)$-distributed random variables, where $LN(\mu,\sigma^2)$ is the log normal ...
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1answer
487 views

Covariance, covariance operator, and covariance function

I am trying to get my head wrapped around this article in Wikipedia. The first definition given there is the covariance of a probability measure $\mathbf{P}$: $$\mathrm{Cov}(x, y) = \int_{H} \langle ...
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2answers
169 views

If $E(Y\mid X)$ is constant then $X, Y$ are uncorrelated.

Last minute studying please tell me how to: Prove that if the expected conditional expected value of the random variable $X$ given the random variable $Y$ - denoted by $E(X\mid Y)$ - is constant ...
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0answers
15 views

Can correlation dimension of an attractor exceed the dimension of the space?

Here is the definition of the correlation dimension: http://en.wikipedia.org/wiki/Correlation_dimension Is there a proof that the correlation dimension cannon exceed the dimension of the space?
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2answers
365 views

Finding a Correlation between Bernoulli Variables?

Let X and Y be Bernoulli random variables. We don't assume independence or identical distribution, but we do assume that all 4 of the following probabilities are nonzero. Let a := P[X = 1, Y = 1], b ...
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3answers
1k views

Is correlation (in some sense) transitive?

If we know that A has some correlation with B ($\rho_{AB}$), and that B has some with C ($\rho_{BC}$), is there something we know to say about the correlation between A and C ($\rho_{AC}$)? Thanks.
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1answer
213 views

Expectation of product of correlated Brownian motions at different time points

Given the information about the correlation of two Brownian motions as $E[dW_1 dW_2] = \rho dt$ and knowing that $E[W_1(t)W_1(t')] = \min(t,t')$, I want to compute $E[W_1(t)W_2(t')]$ I interpret ...
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1answer
37 views

Correlation of two Binomial RVs

Suppose a coin is flipped 30 times. Let X = #heads in first 20 flips, Y = #heads in second 20 flips. I want to find Corr(X, Y). I am only confused on how to find Cov( X, Y) = E[ XY] - E[ X]E[ Y], ...
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1answer
40 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
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1answer
57 views

Compute for Cov(X,Y) and Correlation(X,Y)

Let $(X, Y)$ be uniform on the half disc $D = \{(x, y) : 0 < y, x2 + y2 < 1\}$. How should I approach this problem. Should I solve double integral with inside goes from $-\sqrt1-x^2$ to ...
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1answer
35 views

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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0answers
87 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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1answer
193 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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1answer
7k 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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0answers
53 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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0answers
58 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 ...