Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.

Events (or random variables) are independent when information on some of them tells you nothing about the probability of occurrence (/ distribution) of the others - their joint probability (/distribution) is the product of their marginal probabilities (/distributions).

Pairwise independence (where say X, Y and Z are each independent when taken two at a time) doesn't guarantee mutual independence.

If two variables are independent, their covariance and their correlation is zero (but the converse doesn't apply - zero correlation or covariance doesn't generally imply independence).

If variables are independent, the variance of their sum is the sum of their variances.

Note: Please do not use this tag to refer to independent variables, use instead.

Reference: Wikipedia - independence

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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?
abc
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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 someone provide one?
Flying pig
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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 than two variables, ${\rm Var}(X_1X_2 \cdots…
damla
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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 probability and/or statistical jargon is tossed around…
user68492
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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 distribution laws of $X$ and $Y$ (least of all…
sds
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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 explanation, not a highly technical one.
Victor
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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 destination, as it decreases the probability that he…
Kyle
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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 front-door from B to A? Because then we should get…
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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"?
Clare Brown
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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 between the sample mean and the sample variance of…
JohnK
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What is the relationship between orthogonal, correlation and independence?

I've read an article saying that when using planned contrasts to find means that are different in an one way ANOVA, constrasts should be orthogonal so that they are uncorrelated and prevent the type I error from being inflated. I don't understand…
Carl Levasseur
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If X and Y are uncorrelated, are X^2 and Y also uncorrelated?

If two random variables $X$ and $Y$ are uncorrelated, can we also know that $X^2$ and $Y$ uncorrelated? My hypothesis is yes. $X, Y$ uncorrelated means $E[XY]=E[X]E[Y]$, or $$ E[XY]=\int xy f_X(x)f_Y(y)dxdy=\int xf_X(x)dx\int…
Vegard Stikbakke
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What does "independent observations" mean?

I'm trying to understand what the assumption of independent observations means. Some definitions are: "Two events are independent if and only if $P(a \cap b) = P(a) * P(b)$." (Statistical Terms Dictionary) "the occurrence of one event doesn't…
RubenGeert
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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: (x1, y1) (x2, y2) ... (xn, yn) we can use distance…
Roman
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With categorical data, can there be clusters without the variables being related?

When trying to explain cluster analyses, it is common for people to misunderstand the process as being related to whether the variables are correlated. One way to get people past that confusion is a plot like this: This clearly displays the…
gung - Reinstate Monica
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