# 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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### Functions of Independent Random Variables

Is the claim that functions of independent random variables are themselves independent, true? I have seen that result often used implicitly in some proofs, for example in the proof of independence ...
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### Why zero correlation does not necessarily imply independence

If two variables have 0 correlation, why are they not necessarily independent? Are zero correlated variables independent under special circumstances ? If possible, I am looking for an intuitive ...
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### What is covariance in plain language?

What is covariance in plain language and how is it linked to the terms dependence, correlation and variance-covariance structure with respect to repeated-measures designs?
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### Variance of product of multiple independent random variables

We know the answer for two independent variables: $${\rm Var}(XY) = E(X^2Y^2) − (E(XY))^2={\rm Var}(X){\rm Var}(Y)+{\rm Var}(X)(E(Y))^2+{\rm Var}(Y)(E(X))^2$$ However, if we take the product of more ...
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### Simple examples of uncorrelated but not independent $X$ and $Y$

Any hard-working student is a counterexample to "all students are lazy". What are some simple counterexamples to "if random variables $X$ and $Y$ are uncorrelated then they are independent"?
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### $X_i, X_j$ independent when $i≠j$, but $X_1, X_2, X_3$ dependent?

I've seen the statement: It's possible that random variables $X_i, X_j$ are independent for $i≠j$, but $X_1, X_2, X_3$ are dependent. I haven't been able to find examples of this though. Any ...
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### Covariance and independence?

I read from my textbook that $\text{cov}(X,Y)=0$ does not guarantee X and Y are independent. But if they are independent, their covariance must be 0. I could not think of any proper example yet; could ...
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### 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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### Does statistical independence mean lack of causation?

Two random variables A and B are statistically independent. That means that in the DAG of the process: $(A {\perp\!\!\!\perp} B)$ and of course $P(A|B)=P(A)$. But does that also mean that there's no ...
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### If random variables are drawn from an identical distribution, why doesn't this guarantee they are independent?

Having read a little about exchangeability, I went back to thinking about the iid condition required for the central limit theorem. It struck me that if two random variables are drawn from an ...
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### How do I test that two continuous variables are independent?

Suppose I have a sample $(X_n,Y_n), n=1..N$ from the joint distribution of $X$ and $Y$. How do I test the hypothesis that $X$ and $Y$ are independent? No assumption is made on the joint or marginal ...
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### Does 10 heads in a row increase the chance of the next toss being a tail?

I assume the following is true: assuming a fair coin, getting 10 heads in a row whilst tossing a coin does not increase the chance of the next coin toss being a tail, no matter what amount of ...
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### Does independence imply conditional independence?

If two or more variables A, B, C, etc. are jointly mutually independent of one another, does this imply that that they are also conditionally independent given some set of conditioning variables X, Y, ...
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### 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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### When two events $A$ and $B$ have no result in common

I and my friends just had a little discussion whether the events are independent or dependent if they have no outcome in common. I thought that they have to be independent. When two events are ...
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### Question about correlations and independence

My professor was discussing correlation and when it implies independence. It was fairly clear to me that, if $X$ and $Y$ are independent, then their correlation is zero. The reverse direction is less ...
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### Are your chances of dying in a plane crash reduced if you fly direct?

I recently had a disagreement with a friend about minimizing the chance of dying in a plane due to a crash. This is a rudimentary statistics question. He stated that he prefers to fly direct to a ...
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### Test for IID sampling

How would you test or check that sampling is IID (Independent and Identically Distributed)? Note that I do not mean Gaussian and Identically Distributed, just IID. And idea that comes to my mind is ...
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### 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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### What is implied by i.i.d.?

It's common to see statements that data must be i.i.d. But if data in a time series are independent, aren't they just noise? At one point I thought, they're only referring to the errors, i.e. the ...
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### Test of independence vs test of homogeneity

I am teaching a basic statistics course and today I will cover the chi-squared test of independence for two categories and the test for homogeneity. These two scenarios are conceptually different, but ...
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Under what conditions does the below statement hold: X and Y are uncorrelated if and only if X and Y are independent. I totally understand that this statement does not always hold, but I would ...
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### Confused about independent probabilities. If a fair coin is flipped 5 times, P(HHHHH) = 0.03125, but P(H | HHHH) = 0.5

I'm confused about how to reconcile the probability of independent events not having anything to do with prior history, but sequences of events do (seemingly) take into account prior history. This ...
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### What's the difference between "mean independent" and independent?

As stated in the Econometrics textbook (Introductory Econometrics by Wooldbridge): When $E(u|x)=E(u)$ holds, we say that $u$ is mean independent of $x$. Why can't we simply say that $u$ is ...
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### Variance of annual return based on variance of monthly return

I'm trying to understand the whole variance/std error thing of a time series of financial returns, and I think I'm stuck. I have a series of monthly stock return data (let's call it $X$), which has ...
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### 10% rule for sample sizes

In an introductory stats book by Nicole Radziwill "Statistics (the easier way) with R", an assumption used for nearly every statistical test (e.g.t-tets, anova, etc) is that the sample size ...
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### Are observations independent in bootstrapped resamples?

The bootstrap is often used for nonparametric inference. However, in some cases, it is useful to bootstrap and then conduct parametric tests within each resample (optionally, see References, but this ...
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Given 3 random variables $X_1$, $X_2$ and $Y$. $Y$ and $X_1$ are independent. $Y$ and $X_2$ are independent. Intuitively I would assume that $Y$ and $X_1+X_2$ are independent. Is this the case, and ...