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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CPD Logical OR - context-specific independence [on hold]

$X_1$ and $X_2$ are the parents of $Y$. $X_1$ and $X_2$ are two independent Bernoulli random variables. $Y = OR(X_1, X_2)$. Are $X_1$ and $X_2$ context-specific independent when $Y = 0$? By knowing ...
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11 views

Conditional independence identity

If A and B are independent and conditionally independent given C, but A and C and B and C are not necessarily independent, then $ P(A,B | C) = P(A | C) P(B | C) $ Is it also true that $ P(C | A, B) ...
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133 views

statistics of airline crashes

Ive been going back and forth with a friend of mine on the issue of airline crash statistics. The concept of "independent variables" seems to be the biggest disagreement between us, so I was hoping to ...
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9 views

Testing whether two samples come from the same population without assuming independence?

I'm doing some analysis of frequency data coming from the Sun. I want to test the null hypothesis of two different signals coming from the Sun having the same frequency. I measured both of these ...
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51 views

Normal distribution and independence

I was reading about white noise and it stated: Although $\varepsilon_t$ & $y_t$ are serially uncorrelated, they are not necessarily serially independent, because they are not necessarily ...
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22 views

Independence betweeen sets of random variables

Given two random variables $A$ and $B$, I know we can call them independent if their joint PDF is factorizable to a product of their marginal PDFs: $f_{A,B}\left(A,B\right)=f_{A}\left(A\right)\cdot ...
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1answer
36 views

The Independence in Independent Component Analysis - Intuitive Explanation

I could use a little bit of help in understanding a concept with regards to ICA: ICA decomposes a multivariate signal into 'independent' components through 1. orthogonal rotation and 2. maximizing ...
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Why isn't independence checked standard in ANOVA?

The standard procedure for a one-way analysis of variance seems to be: fractile-diagram (or Q-Q plot) for checking normality Bartlet's test for heteroscedasticity But as standard I have never ...
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Independence of iid samples when function is applied to to one member of sample [duplicate]

I have tuples, $\{(X_i,Y_i)\}_{i=1}^n$, which are iid. Suppose I introduce a new variable, $T_i = g(X_i)$, where the function $g$ is arbitrary. Is it true that $\{(T_i, Y_i)\}_{i=1}^n$ are iid as ...
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52 views

If $X$ and $Y$ are normally distributed random variables, what kind of distribution their sum follows?

I was reading this question. It is about notation but I would like to ask something regarding the sum of two normally distributed random variables. If $X$ is a normally distributed random variable ...
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1answer
28 views

Conditional Independence vs. Collinearity

I know that when two predictor variables are correlated then that increases error and distorts the results, which is called multicollinearity. But what is conditional independence, is it the same as ...
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26 views

How to find this conditional probability in sampling without replacement in this experiment?

My problem is the following: I have an urn with 10 different names. Consider a particular name (ex. John). I want to estimate: 1) the probability to extract John 2) the probability to extract John ...
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2answers
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How do you reconcile the definitions of multiple mutually independent events and multiple mutually independent random variables?

1) If $A, B, C$ are events then to prove mutual independence it's not enough to show $P(A,B,C)=P(A)P(B)P(C)$. You also have to show $P(A,B)=P(A)P(B)$ etc. 2) For random variables, the definition of ...
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24 views

Test for differences in frequencies of categorical variable with multiple responses

Background Interviews with two groups of therapists were conducted: experienced therapists and inexperienced therapists. Each statement of the interview was classified into 9 different categories. ...
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Independence of sexes of children born to a couple

Suppose $100p\%$ of babies born are male. Sometimes one reads that $p\approx 0.505$ or something like that. If a couple has $n$ children, it is often assumed that the sex of each child is ...
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31 views

Independence between sibling groups?

I am a novice at statistics and I would sincerely appreciate your help with my question: I am working with a control group and 2 sibling groups and have a mixed design (i.e., one withing group ...
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14 views

Continuous Independence Test using only sufficient statistics

does anybody know a way to test wether two continuous variables X and Y are independent only using their sufficient statistics? The work I have seen so far was assuming that N values of variable X and ...
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Can I use the Chi squared test to see if my data fit to the Wigner-Dyson distribution indicating chaotic behaviour?

I recently read the paper about Quantum Chaos in Feshbach resonance spectra of ultracold Erbium [1]. The authors analyze their data statistically using the random matrix theory framework for ...
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Does this mixed model violate assumptions of independence?

A disturbance event caused damage to 5 streams (Set1). To quantify this damage, five additional unimpacted streams (Set2) were picked for comparison. During the selection process every effort was ...
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Generating pairwise independent bits

Following the algorithm below, it's possible to generate $2^N-1$ pairwise independent bits from N independent random bits: If I am using these pairwise independent bits to derandomize an algorithm ...
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1answer
35 views

Dependence of Error: Does it matter for data-driven models?

Linear regression assumes that the errors of the response variable are independent of each other. Lets assume that a data-driven model like a random forest or multi-layer perceptron is trained/formed ...
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40 views

Conditional independence conjecture

Suppose there are four random variables (events), $A$, $B$, $C$ and $X$. If we have $X\perp A|B$, saying $X$ and $A$ are conditional independent given $B$, then I want to ask that whether we have ...
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22 views

Independence test for two small, exhaustive, categorical variables

I've got a categorical variable $var$ and a binary variable $critere$, from a pretty small (n = 300) but exhaustive dataset (i.e., the dataset contains the whole population that I want to study). I ...
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34 views

Does the degree to which events are independent increase as the number of events increase?

I'm considering a problem of independence of events. Specifically, I am considering whether events are more independent the greater the number of events. Two examples might help. Example 1: There ...
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16 views

how to quantify the dependence between two sparse signals?

Hi and thank you for your help. I am happy to provide more context/details. In my research, I have a system that has many sensors X_1..X_N and using this ...
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268 views

Data augmentation techniques for general datasets?

In many machine learning applications, the so called data augmentation methods have allowed building better models. For example, assume a training set of $100$ images of cats and dogs. By rotating, ...
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When used for feature selection, does the chi-squared test require the features to be nonnegative?

scikit-learn says chi squared test used for feature selection in classification problems and implemented by sklearn.feature_selection.chi2 requires the feature ...
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45 views

Does rank of observation matrix tell anything useful when applying machine learning?

Suppose I have an observation matrix of size $N \times M$ where $N$ is the number of samples and $M$ is the number of variables. If the rank of the observation matrix is $R<M$, does it tell ...
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96 views

What kind of feature selection can Chi square test be used for?

Here I am asking about what others commonly do to use chi squared test for feature selection wrt outcome in supervised learning. If I understand correctly, do they test the independence between ...
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13 views

How do I account for non-independence in occupancy modeling?

I have conducted a survey to look at why badgers are visiting some gardens but not others. I have data on badger presence / absence and garden features that might influence badger presence in that ...
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87 views

Consequence of violation of independence assumption on estimates of standard errors

from the first chapter , Introduction to Multilevel Analysis , p.5 of the book , it is written that : Standard statistical tests lean heavily on the assumption of independence of the ...
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Suppose $X_1 X_2, …, X_n$ are $n$ independent variables, is their Covariance matrix, $\Sigma$, diagonal?

Suppose I have $n$ variables $X: X_1, X_2, ..., X_n$ that are independent from each other. Which means that: if $i≠j$, then $\text{Cov}(X_i, X_j) = 0$ As a consequence, I'm wondering if their ...
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29 views

Solving the probability of independent events without the complement

Suppose that virus transmission in 500 acts of intercourse are mutually independent events and that the probability of transmission in any one act is $\frac{1}{500}$. What is the probability of ...
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41 views

Independent RVs theorem: rigorous?

I am reproducing here theorem (#3.30) from "All of Statistics" by Larry Wasserman: Let X and Y have joint pdf $f_{X,Y}$ . Then $X\perp Y$ if and only if $f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y)$ for all ...
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58 views

Are two variables that are dependent also correlated? / Can two dependent variables not be correlated? [duplicate]

I cannot find the answer to my question anywhere, I hope someone can help. I did a Chi-Square test with several variables to see if they were dependent or independent. The result showed that they ...
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22 views

Data Assumptions for AIC model comparisons

I recently started digging into statistical information criteria, more specifically the Akaike Information Criterion. As the literature I have read so far does not cover this, I was wondering whether ...
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47 views

Are Latin hypercube samples uncorrelated

I understand the basics to Latin hypercube sampling, such as implemented by the algorithm LHSA mentioned in the book Design and Modeling for Computer Experiments. But I'd like to make sure: 1, n ...
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16 views

statistically analyzing independent variables

I am kind of new to statistics. I have 4 independent variables that has been observed from a system in 4 different configurations. At this point, I don't know what are the best statistical functions ...
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60 views

importance of independence among random variables

I always read random variables as being independent and identically distributed. I understand the concept of being identically distributed, because if different random variables are distributed in ...
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45 views

CDF of sum of 3 independent discrete uniform random variables on {1,2,…,n}

What is an approximate closed formula for this probability, with a derivation: p(k,n) is the probability, that among $n$ PC discs and $k$ errors in sum on them, there will be at least $1$ disc ...
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41 views

assuming independency between independent variables in multiple regression?

I heard that multiple regression assumes that the independent variables are correlated somehow. So when we convert the multiple regression into SEM diagram, we see covariance arrows are drawn ...
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52 views

Are $\hat{\beta}_{\text{ls}}$ and $S^2$ independent if errors are not normally distributed?

When estimating a linear model $$ Y_i = X_i\beta + \varepsilon_i \quad \quad 1\leq i\leq n$$ We have $\hat{\beta}$ the least squares estimation of the slope and the estimation of the variance, $S^2 = ...
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50 views

Show that $Y_1 X_1 + Y_2 X_2$ $\,{\buildrel d \over =}\,$ $(Y_1^2+Y_2^2)^{1/2}X_1$

I would like verification of my solution to the following problem. QUESTION: Let $X_1, X_2 \,{\buildrel iid \over \sim }\, N(0,1) $ and let $Y_1, Y_2$ be two independent random variables ($X_1, ...
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33 views

Test for independence in 3x2x2 table with small expected cell values

I have three unordered categorical variables. I have made a 3x2x2 table, but I'm uncertain of how to test for independence. The usual manner is a chi-squared statistic, but I think this is ...
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121 views

What is the demonstration of the variance of the difference of two dependent variables?

I know that the variance of the difference of two independent variables is the sum of variances, and I can prove it. I want to know where the covariance goes in the other case.
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22 views

How to run ANOVA on multiple groups of samples, each composed by different variables

I have a $m$ x $n$ matrix, where the $n$ columns are split into multiple classes. If I had only a $1$ x $n$ vector, I would have used ANOVA to evaluate if all subset of columns had the same ...
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4 views

Chi-square test for independence of categorical variables [duplicate]

I'd like to know about independence of categorical variables. I read an article from https://explorable.com/significance-test-2 H0: “Having breakfast in the morning has no effect on the grades of ...
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2 views

Comparing change in lateral root densities between treatments for different genotypes

I have been growing 4 genotypes of the same plant and been recording their lateral root density under both control conditions and a salt treatment. I have calculated the decrease in lateral root ...
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102 views

Random variables independence

I need to check if $Z$ and $W$ are dependent or not. $X,Y \sim \mathrm{Exp}(2)$ Then I define: $Z=X-Y \ \text{,}\ \ W=X+Y$. Now How I can check that $Z$ and $W$ are dependent or not ? I know ...
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In linear regression, what does $\beta_1 = 0$ really mean?

If granted omniscience and we know that $\beta_1$ in a multiple linear regression model is truly 0, what does that mean in words (and math notation)? The model is: $Y = \beta_0 + \beta_1X_1 + ...