Events (or random variables) are independent when information on some of them tells you nothing about the probability of occurrence (/ distribution) of the others. Please DO NOT use this tag for independent variable use [predictor] instead.

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sum of dependent random variables [on hold]

I want to find a function that determine connection between x and y that x and y both of them are determine by a common other factor. in fact x and y are dependent variables. function is like a ...
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1answer
27 views

Point-Biserial Correlation (r) for non homogeneous independent samples

I am performing an independent t-test, in which the independent variable is the "group" which has two values A and B representing an approach the participants used, and the dependent variable is a ...
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1answer
45 views

ANOVA - Assumption of Independence

I am also a little bit confused about the assumption of independence. I have the following situation: one continuous dependent variable (radiant power of a medical device) a few independent ...
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0answers
20 views

Is Independent jeffreys prior different from independent reference prior?

I have a model involving two scalar parameters $\theta_1$ and $\theta_2$ and derive the Jeffreys prior for $\theta_1$ and $\theta_2$ independently (so for, e.g. $\pi(\theta_1)$, setting in the ...
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1answer
64 views

Independence of variables

Q:Two variables X and Y have same mean and variance.If U=X+Y and V=X-Y then, are U and V independent and correlated? I found that U and V are uncorrelated. But don't know how to check for ...
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2answers
51 views

Estimate point in metric variable where probability of success of a conditionally Bernoulli distributed variable changes (independent observations)

There is a metric variable X and a conditionally Bernoully distributed variable Y, where the probability of success of Y changes at a threshold x of variable X. The obervations are independent. I want ...
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0answers
19 views

Can longitudinal studies ever be treated as having independent observations?

This may be a dumb questions, but for sake of statistical analyses I'm wondering if the data of a longitudinal study can ever pass the assumption of having independent observations? For example, I am ...
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0answers
20 views

Practicality of sparse inverse covariance matrix assumptions

For a set of $p$ datapoints in $m$ dimensional space, if the features are packed in a $p\times m$ matrix $X$, then $C = XX^T$ is the covariance matrix and $K = C^{-1}$ is the inverse covariance ...
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1answer
38 views

Non Linear Endogeneity

Consider the following Linear Regression Model.$$y_{it}=x_{it}\beta+\upsilon_{it}$$ where x is a scalar and $$cov(x_{it},\upsilon_{it})=0$$ It is know, however, that $cov(.)$ is a measure of ...
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1answer
32 views

Independent and conditionally independent

I was wondering if two variables can be independent and conditionally independent. For example, A and D are independent. But are they also independent given the evidence C? I think they are, because ...
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1answer
57 views

If $P(u,v|x,y)=P(u|x)P(v|x)$, does it follow that $P(x,y|u,v) = P(x|u)P(y|v)$?

Let $u,v,x,y$ be four random variables such that: $$P(u,v|x,y)=P(u|x)P(v|x)$$ The question is: Does it follow that $$P(x,y|u,v) = P(x|u)P(y|v)$$?
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Chi-squared test of independence for biased data

I'm working with a survey dataset consisting of 28807 observations (8470 males and 20337 females). I'm trying to determine the association between dichotomous variables, for instance, sex (Male, ...
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0answers
17 views

Test of independence for categorical and discrete variables

First time writing question on stack. Mostly, I could find the answer needed from previous questions. This time couldn't find such a. Lets say I have categorical variable y(dependent) which can have ...
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3answers
477 views

Statistical Independence in the real world

I read the following article about statistical independence. In summary, the article argues that "It is time for science to retire the fiction of statistical independence," and goes on to explain ...
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1answer
32 views

Why can a process with independent increments never be a stationary process?

Why can a process with independent increments never be a stationary process? I don't understand the reasoning behind this. Thanks !
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1answer
22 views

Conditions for groups in independent samples test

I wish to conduct an experiment where I have a number of different groups, let's say four groups of X people each. Each person in each group partakes in a similar experiment, with each group testing a ...
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0answers
32 views

Correlation between two explanatory variables

I have a multiple regression. Two of my independent variables are "Repeated Partnerships" and "Company size". When adding the explanatory variable "Company size" to the regression it is statistically ...
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0answers
32 views

Is the limit of n-wise independent events independent?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. ...
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0answers
9 views

Mutual information on only a subset of samples

Suppose I have $M$ sample sets, each of size $M_i$ with each sample being the observation $(X_i^j,Y_i^j)$. Each sample set is drawn from a different source so they may have source specific noise. I ...
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2answers
565 views

For intuition, what are some real life examples of uncorrelated but dependent random variables?

In explaining why uncorrelated does not imply independent, there are several examples that involve a bunch of random variables, but they all seem so abstract: 1 2 3 4. This answer seems to make ...
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1answer
31 views

Some details about the Box-Muller transform method

I am confused about why $Z_0$ and $Z_1$ are independent. It seems like they both rely on $U_1$ and $U_2$. Could someone prove the statement?
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1answer
177 views

Conjecture related to Kolmogorov 0-1 Law (for events)

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1$. ...
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0answers
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Does having a significant result from chi-squared test for independence also means that the odds ratio is significantly different from 1?

Suppose I have 2 by 2 contingency table. I performed Pearson's chi-squared test for independence and the results are significant. Does that mean the odds ratio that I calculated based on the 2 by 2 ...
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21 views

Does value difference metric assume independence?

Value difference metric is a distance measure for nominal attributes. It is used in classification problems. According to this metric, the distance between two points $x$ and $y$ is calculated as ...
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1answer
37 views

Is there a central limit theorem for i.n.i.d. variables when normalised by inconsistent variance estimate?

I am wondering whether there exists a central limit theorem for the following situation. Consider the sum of normally distributed variables $\epsilon_i$ with unequal variances according to ...
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1answer
57 views

How do we prove these $\sigma$-algebras are not independent?

Given the sequence $(X_n), n=1,2,... $, of iid exponential random variables with parameter $1$, define: $$ M_n := \max \left\{ X_1, \frac{X_1+X_2}{2}, ...,\frac{X_1+\dots+X_n}{n} \right\} $$ I want ...
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1answer
44 views

Correlations between 4 variables

I have $m_{1R}$, $m_{2R}$, $m_{1L}$, $m_{2L}$, and about them, I know that $Corr(m_{1R}, m_{1L}) = 0$ $Corr(m_{2R}, m_{2L}) = 0$ $Corr(m_{1R}, m_{2L}) = 0$ $Corr(m_{2R}, m_{1L}) = 0$ Suppose I ...
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0answers
19 views

How can independence be represented efficiently?

Consider a probability distribution over a high dimensional space. We would like to find an encoding describing the distribution, so that we can approximately compute the expected value of most random ...
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0answers
12 views

Two Sample test of proportions with dependent trials

I have data $$X_1,\ldots,X_T \qquad Y_1,\ldots,Y_T$$ where $X_i \sim \text{Categorical}(a_1,\ldots,a_k)$ and $Y_i \sim \text{Categorical}(b_1,\ldots,b_k)$, but the $X_i$ are not independent of one ...
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1answer
17 views

analysis of variance - independence of samples and independence of individuals

I am a bit confused with the concept of "independence of samples" in Analysis of Variance methods. One condition for performing an ANOVA is apparently "independence of the samples". Most of my ...
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2answers
69 views

Some elements dependent, but collectively independent?

Is it possible that some collection could be independent, even if some of its elements were dependent? Or is collection always independent iff its elements are independent? Or perhaps this calls for ...
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1answer
27 views

Independent and dependent gives the same results?

Is there any situations where considering two random variables independently and dependently leads to same computational result ? Is there a reason why this might happen ? My specific problem deals ...
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929 views

The paradox of i.i.d. data (at least for me)

As far as my aggregate (and scarce) knowledge on statistics permits, I understood that if $X_1, X_2,..., X_n$ are i.i.d. random variables, then as the term implies they are independent and identically ...
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1answer
149 views

Independence of $\min(X,Y)$ and $\max(X,Y)$ for independent $X$, $Y$?

What's the reasoning for checking the independence of $$\min(X,Y)$$ and $$\max(X,Y)$$ for independent r.v.s $X,Y$? Is it possible that $\min$ and $\max$ both select the same r.v. in which case they ...
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31 views

Independence of two white noise processes

Say I have two white noise processes $e_{t}\sim WN\left(0,\,\sigma^{2}\right)$ and $u_{t}\sim WN\left(0,\,\sigma^{2}\right)$. Can I say that these two processes are independent or do I need to make ...
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1answer
36 views

The books always give the caveat “if independent”

I'm looking for statistics and probability books/resources that tackle dependant variables/events/etc. The real world is almost never iid. I'm interested in the mathematics of solving problems that ...
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70 views

Prove that two random variables are independent in this specific case

Suppose we have three independent ($N \times 1)$ complex vectors $\mathbf{w}$, $\mathbf{e}$ and $\mathbf{f}$ be . Assume that $\mathbf{f}$ and $\mathbf{w}$ are of unit norm and isotropically ...
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34 views

Independence and the normal distribution

You know those times that the professor assigns a problem, not to test your knowledge of math, but to test your deep understanding of the material? Well that's what I'm dealing with right now. ...
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19 views

Is there a name for this simple probability relation? [duplicate]

Namely: $\quad \frac{P(A \cap B)}{P(A) P(B)}$ Cleary this is 1 if A and B are independent, and I think it would be 1/2 if A = B. So it seems like this is some measurement of dependence. Is it ...
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2answers
196 views

Understanding distance correlation computations

As far as I understood, distance correlation is a robust and universal way to check if there is a relation between two numeric variables. For example, if we have a set of pairs of numbers: ...
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1answer
18 views

How to check if there is a dependency of a numeric variable on other numeric variables?

I need to check if a numeric variable $y$ depends on a set of other numeric variables $(x_1, x_2, ..., x_n)$. I do not know anything about the form of possible dependency (I do not even know if it ...
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0answers
6 views

Independence of quadratic forms of random variables [duplicate]

If $X$ and $Y$ are two vectors of random variables, such that $X$ and $Y$ are independent, is it true that $X^TX$ and $Y^TY$ are independent? If yes, how do we prove it?
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1answer
27 views

Pairwise vs. Mutual independence for normal random vector

Lets say that I have a vector $X$ for which the marginal distribution of each element $x_1,x_2,...$ is normal with variance = 1. Additionally, assume that $X$ is pairwise independent. Does this ...
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1answer
121 views

What does $\mathbb{P}(A \cap B) > 0$ tell?

This is related to the conditional chain rule in conditional probability: If for two sets of a probability space I've been given: $$\mathbb{P}(A \cap B) > 0$$ Then what does this mean? Is it ...
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30 views

Simultaneous confidence interval with $\overline{X}=0$

Suppose that $\overline{X}=0$ and that linear regression model is valid. Show that $\hat{B_0}$ and $\hat{B_1}$ are independent. What can you conclude about the construction of simultaneous ...
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Is the independence assumption broken when collecting data from two people who interacted with each other?

If data was collected from two participants who were interacting in an experiment, would this be breaking the independence assumption?
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58 views

Contraction between Pearson and Pairwise.prop.test

In R I have two vectors a and b with same length. The vectors contains number of times a game has been played. So for example ...
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1answer
50 views

Covariance between $Z$ and $Y$ where $Z=XY$

What is: $${\rm Cov}(Z,X)$$ when $Z = XY$ and $X$ and $Y$ are independent? In essence, is the product of two independent variables independent of either one?
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2answers
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Is or when is $\mathbb{P}(X_1 \in A_1, … X_n \in A_n)$ equivalent to $\mathbb{P}(A_1 \bigcap … \bigcap A_n)$?

In the context of independence: Is $\mathbb{P}(X_1 \in A_1, ... X_n \in A_n)$ equivalent to $\mathbb{P}(A_1 \bigcap ... \bigcap A_n)$? $X_i$s are random variables, $A_i \subset \Omega$ (the sample ...
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1answer
10 views

How can model misfit introduce dependency?

I read somewhere recently that mis-fitting a regression model can introduce dependency into the model. Unfortunately I cannot find the where I read this. Can anyone explain how mis-fitting a ...