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Questions tagged [independence]

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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Do I discard all my dependent variables as proved by chi-squared test of independence?

I have 134 categorical columns in my data. 7 of which are categorical variables [ one variable is highly unbalanced and has 34 classes while all other variables just has 3-5 classes in each variable ...
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What are the differences between different variable dependency measures? [on hold]

There are different measures for calculating dependence among variables (see here). For example, Pearson's coefficient which computes the correlation between variables and is most useful in detecting ...
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Why does independence imply zero correlation?

First of all, I'm not asking this: Why does zero correlation not imply independence? This is addressed (rather nicely) here: https://math.stackexchange.com/questions/444408/why-does-zero-correlation-...
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1answer
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$X\perp Y \Leftrightarrow Y\perp r(X)$?

Consider the random variables $Y$ and $X$. Let $\mathcal{X}$ denote the support of $X$. Let $\mathcal{Y}$ denote the support of $Y$. Let $r:\mathcal{X}\rightarrow \mathbb{R}$. I have doubts about the ...
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Co-training independent views

I found a tutorial on co-training that mentions that I need two views for training, which are conditionally independent. Could someone explain to me what this means regarding datasets?
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How to Use Chi-Squared Test for Inference about Three-way independence

If I recall correctly, three random variables X, Y, and Z are three-way independent iff these two statements are met: P(X∩Y∩Z) = P(X)P(Y)P(Z) X, Y, and Z are all pairwise independent of each other. ...
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23 views

I.i.d.-ness of some functions of random variables

I have some doubts on the i.i.d.'ness of some functions of random variables. The framework What I'm describing below is a simplied version of a well known model in economics of demand and supply. ...
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1answer
26 views

Probability of event independent of random variables [closed]

Let {$X_n$} be a sequence of independent random variables and for each $n$, the event $A$ is independent of $X_1$, $X_2$,..., $X_n$. Show that $P(A)$ is $0$ or $1$.
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1answer
27 views

Finding distribution and checking independence of transformed normal variables

$X,Y,Z$ are three independent random variables following standard normal distribution. Consider a real function $f$ such that \begin{align}f(x)&=1 , x\geq 0 \\ &= -1, x<0\\ \end{align} Let $...
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1answer
49 views

Identify dependent or independent blocks of time series (clusters)

maybe I am lost in translation but I need your help. Description: Having long time series of two variables I create some blocks (or clusters) with the method of peak over threshold but I need to ...
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1answer
35 views

Independence between sum of $F(X_i)$ and the cumulative distribution function $F$

I am stuck in the following problem. $X_1,\ldots,X_n$ is a random sample from a continuous distribution with Cumulative Distribution Function (CDF) $F$. Prove that the distribution of $T=\sum_{i=1}^m ...
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2answers
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Can independent/dependent events be thought of graphically?

My understanding is that events are subsets of the total outcomes in a sample space. So if two events are mutually exclusive, then they (the sets) do not overlap in the sample space. This can be seen ...
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Measuring effects of IV (constant) on DV

I have an independent variable (a bias) that is technically a constant. I want to measure the effects of the IV on the DV. How do I do this? I'm using SPSS. Let's say, participants produced A or B ...
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2answers
33 views

Covariance of two normally distributed variables

I saw in a statistic book that "It can be prooved that if two normally distributed variables have covariance = 0, they are independent". How can I start this proof? Can I say that $cov(X,Y) = E(XY) ...
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0answers
35 views

Can two random variables be independent in some basis and dependant in other?

If some random variables forming N dimensions are dependant on each other is it possible that in a different coordinate system they'd be independent? For example if (X, Y) are two dependent RVs is it ...
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14 views

Use of a predictor variable for a dependent variable that is directly related to the predictor variable

If I would like to predict a binary variable x, and x is true if y is true and ...
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1answer
17 views

independence of product of independent random variables with one of the variables

Take random variables A and B, which are independent. Is it the case that AB (product) is independent of B?
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i.i.d. random vectors [duplicate]

If $(X_1,Y_1), (X_2,Y_2)$ are independent random vectors having the same joint distribution function $F$, then is it correct to say: $E(X_1)=E(X_2)$ and $E(Y_1)=E(Y_2)$ (the same for variance); Both, ...
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1answer
11 views

Relationship/independence between 2 binary varibales

I got a dataset in Excel with 2 dichotomous variables: trained (1 or 0), part-time (1 or 0). The question asks if whether there is a relationship between those who have been trained and whether they ...
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0answers
19 views

F ratio independence of numerator and denominator

For a particular hypothesis testing problem I have the quantity $F = \frac{\sum_{j=1}^{k} (\overline{X_j})^2}{\sum_{j=1}^{k} \sum_{i=1}^{n} (X_{ji} - \overline{X_j})^2}$ where $X_{ji}$ is the $i^{th}$ ...
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0answers
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Can I take a random sample of my very large data set to overcome non-independence?

I am trying to run a regression model on a very large time series data set (comparing flow noise to vehicle speed, pitch and dive state). Because my samples are taken about every minute (with some ...
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1answer
49 views

When knowledge of one event tells us nothing about another

I have an example; Say I have two genes on two different locations on two different chromosomes. P(A) = gene is from location 1 P(B) = gene is from location 2 Geneticists believe that knowledge of ...
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0answers
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Confused about multilevel analysis and non independence of observations

I'm still struggling with my understanding of multilevel analysis, wondering if it applies or not to my problem. I'v read here the following (where author gives an example of a multilevel model with ...
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0answers
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Independence test with priors

Say you are selling a product, and you know from experience that the green version of the product sells better than the blue version. But you have two types of customer A and B, and you want to know ...
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2answers
29 views

What is the difference between having input parameters that are independent/dependent in ML tasks?

This is more of a general question. But, when you feed in inputs into a machine learning algorithm, are the inputs typically dependent or dependent? What are the implications if the inputs are ...
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Independence in this Bayes net

Consider this Bayes net A,B,C forms a v-structure. $B \not\!\perp\!\!\!\perp C | A$, B is not independent with C if A is observed. My question is, if B,A,D are all given, can we write $p(C|B,A,D) = p(...
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1answer
18 views

Determining Conditional Independence from Marginal Independence?

so if I have a 3 columns of binary variables X, Y and Z with their respective values and I would like to determine whether X,Y are conditionally independent given Z. How can I go about doing this? ...
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1answer
30 views

How do you verify independence of a pair of uniform random variables in MATLAB? [closed]

If $A$ is a subset of $R$ and $X$ is a random variable. I have two variables $X_1$ and $X_2$. $I$ being $1$ if $X$ in subset $A$, and $0$ if not in $A$. Let $U$~$U(0;1)$ and determine if this pair ...
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0answers
27 views

When does $E[f(X_i)]=E[f(X_j)], i\neq j$?

Suppose we have random variables $X_1, \dots, X_N$, with joint probability distribution $F_{X_1,\dots,X_N}$. Under what conditions does the following equality holds? $$E[f(X_i)]=E[f(X_j)],\ \ i\neq ...
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2answers
45 views

How to find out if there is any real pattern in the data set?

Let's assume that we have a regression problem (in the machine learning sense). Our data set consists of pairs of features vectors and numeric targets. It might be the case that there is absolutely ...
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1answer
26 views

Interpreting the Chi Square for a 2x2 contingency table

Here is my problem. I need to implement an algorithm to find if there is a dependency between 2 categorical variables. My population can vary a lot as it is an input I don't have control over. My ...
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0answers
9 views

Spherical distribution counterexample

Let $X \in \mathbb{R}^{n \times p}$ be a random matrix. The distribution of $X$, call it $F(X)$, is said to left-spherical if for any orthogonal matrix $O$ such that $O'O = I_n$, $F(OX) = F(X)$. ...
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1answer
357 views

Sums of normal random variables

Consider a sample of n independent normal rvs. I would like to identify a systematic way of calculating the probability of having the sum of a subset of them larger than the sum of the rest of rvs. An ...
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2answers
50 views

10% rule for sample sizes

In an introductory stats book by Nicole Radziwell "Statistics the easy way with R" , an assumption used for nearly every statistical test (e.g.t-tets, anova, etc) is that the sample size should not ...
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1answer
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How can observations of random variables be IID, if they are not themselves random variables?

Suppose we perform some experiment which results in an outcome $\omega \in \Omega$. A random variable $X$ maps $\omega$ to a real number, and the (discrete) distribution $P(X)$ maps $X$ to $[0, 1]$. ...
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202 views

Interpretation of low intraclass correlation coefficient (ICC) values and assumptions of independence

I have an unbalanced repeated measures data set and my main goal is to see what stats are correlated with win percentage. There is an unbalanced number of observations per player. With my data set I ...
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3answers
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Is the sample correlation always positively correlated with the sample variance?

The sample correlation $r$ and the sample standard deviation of $X$ (call it $s_X$) seem to be positively correlated if I simulate bivariate normal $X$, $Y$ with a positive true correlation (and seem ...
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1answer
23 views

Conditional independence and joint distributions in graphical models

I'm reading Deep Learning by Ian Goodfellow and Yoshua Bengio and Aaron Courville. In chapter 3 about graphical models, to reduce the model complexity, we assume that certain conditional independence ...
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0answers
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Why dependencies would cancel while paramterizing CPD?

Consider that in CPTs of a Bayes Net(structure is fixed) one dependence is fix(parameters are set for $P(X|Y)$) and we are parameterizing other CPDs, is it possible that the other parameters would ...
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0answers
20 views

Conditional dependence between two ordinal variables

I really hope that this question was not answered in any way before, but I couldn't find an appropriate solution to my problem - most probably due to my lack of statistical knowledge. I am trying to ...
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1answer
14 views

Setting boundaries for calculating $P(Y/X>2)$ choosing $dx/dy$ order [duplicate]

Given two independent variables $X$ and $Y$, with marginal pdfs $f_X(x)=2x, 0 \le x \le 1$ and $f_Y(y)=1, 0 \le y \le 1$, calculate $P(\frac{Y}{X} > 2)$. So this can be written as $P(Y>2X)$, ...
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0answers
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Proof of Berkson's Paradox

I'd be very thankful if someone could help me with the proof of Berkson's Paradox. I found this quite helpful thread which I understand How to prove Berkson's Fallacy?. But I'm actually trying to ...
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1answer
35 views

mixed effect model?

I have 365 days of bike sharing demand data for 15 stations. I am thinking of taking each day data as a data point (n=365*15=5475) and relate daily weather variable as well as land use variable. The ...
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3answers
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Does $\mathbb{P}(XY=a)=\mathbb{P}(X=a)\mathbb{P}(Y=a)$ for $X,Y$ id?

Does $\mathbb{P}(XY=a)=\mathbb{P}(X=a)\mathbb{P}(Y=a)$ for $X,Y$ id? What confuses me is that I only find this independence result for $\mathbb{P}(X=a,Y=a)$ i.e. $\mathbb{P}(X=a \cap Y=a)$
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1answer
421 views

Relation between independence and correlation of uniform random variables

My question is fairly simple: let $X$ and $Y$ be two uncorrelated uniform random variables on $[-1,1]$. Are they independent? I was under the impression that two random, uncorrelated variables are ...
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1answer
122 views

Is the Chi-Square Test of Independence the best option for 3x2 contingency table?

I have some data which in the terms of a 3x2 contingency table look in the following way. ...
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0answers
29 views

The consequences of ignoring autocorrelation of errors for the LASSO estimator?

In ordinary linear regression, Y = X$\beta$ + $\epsilon$, if the error is autocorrelated, then the assumptions under the Gauss-Markov theorem are violated. For example, autocorrelation violates the ...
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0answers
41 views

Converting a sequence of dependent random variables to an iid sequence

I have a sequence $L = X_1, X_2, ..., X_m$ of iid random variables and another sequence $R = X_{m+1}, X_{m+2}, ..., X_{n}$ of iid random variables such that each $X_i$, for $1 \leq i \leq m$, is ...
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0answers
29 views

Covariance of two uncorrelated variables multiplied with the same random variable [duplicate]

I have a problem where I am faced with the term $\operatorname{Cov}[XY,XZ]$. However, I do not know what to do with this term. I may assume that $Y$ and $Z$ are independent and that $X$ is independent ...
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1answer
34 views

Implications of i.i.d. sample

I have the following question: I have managed to solve it but I wasn't sure if my reasoning was correct. So I can express the OLS estimator as $\sqrt{n}(\hat{\beta} - \beta) = (\frac{1}{n}\sum_{=1}^{...