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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Two random variables are dependent given a third random variable, and independent otherwise

Suppose that $X$ and $Y$ are dependent given $Z$, and independent when not given $Z$. Does this mean that: $$ p(x,y) = p(x) \cdot p(y) \\ p(x,y|z) \neq p(x|z) \cdot p(y|z) $$ Also, are there any real-world examples of this scenario?
mhdadk
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Is the sum of two variables independent of a third variable, if they are so on their own?

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 how can I prove it formally?
matthiash
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How problematic is it to control for non-independent covariates in an observational (i.e., non-randomized) study?

Miller and Chapman (2001) argue that it is absolutely inappropriate to control for non-independent covariates that are related to both the independent and dependent variables in an observational (non-randomized) study - even though this is routinely…
Patrick
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Recommendations - or best practices - for analyzing non-independent data. Specific example relating to pain perception data provided

I am seeking recommendations and/or best practices for analyzing non-independent data. In particular, I am curious about non-independent data that does not reflect typical repeated-measures time-based data in which data for the same question(s) or…
Patrick
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Using data from both eyes in between group comparisons

The research question: Do vision test results differ among 3 groups (of eyes) separated on the basis of retinal health: Normal: Mild Disease; Moderate disease. Problem: Both eyes of each person are tested and we would like to use data from both…
MES
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How is mean independence defined?

Suppose $X$ and $Y$ are random variables. As I understood it, mean independence is defined as follows $Y$ is mean-independent of $X$ iff $\Bbb E[Y|X] = \Bbb E[Y]$ But my professor in class gave the following definition in class yesterday $Y$ is…
Stan Shunpike
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Are the products of dependent and independent random variables independent?

Let A, B, C, and D be four random variables such that A and B are independent, and C and D are dependent. It is unknown whether A and C are independent nor whether B and D are independent. Let E and F represent the products E = AC and F = BD. Are E…
Tyler Streeter
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Correlated/dependent random variables

This may be a very basic question or a very difficult one, I am not sure. What does it mean if two random variables are perfectly correlated? What does it mean if two random variables are perfectly dependent (whatever that means)? When two random…
Cagdas Ozgenc
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Are two different ways of measuring an individual statistically indepent?

I'm measuring genetic diversity from every individual in a sample using two different types of genetic information. These types of genetic information have different properties, and mutate independently of one another, yet will always be inherited…
anjama
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Eliminating statistical dependency from a set of random variables

Let's say I have a finite set of binary random variables. Does there exist, for any such set, another finite set of binary random variables that carries approximately the same information, but whose members are all statistically independent of each…
user124644
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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 exists). So, my goal is to check if there is…
Roman
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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 and realize that I need additional bits, is it…
Neal Jean
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Mutual Independence Definition Clarification

Let $Y_1, Y_2, ..., Y_n$ be iid random variables and $B_1, B_2, ..., B_n$ be Borel sets. It follows that $P(\bigcap_{i=1}^{n} (Y_i \in B_i)) = \Pi_{i=1}^{n} P(Y_i \in B_i)$...I think? If so, does the converse hold true? My Stochastic Calculus…
BCLC
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Example of Darmois Skitovitch theorem

In looking at something, I stumbled across the awesome Darmois-Skitovitch theorem: Define two random variables $y_1$ and $y_2$ as linear combinations of independent random variables $s_i$ $(i=1, · · · , q)$: $$y_1=\sum_{i=1}^q\alpha_i…
roundsquare
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Can pieces of evidence that arise from different aspects of the same phenomenon be independent?

Say I want to know whether my roommate is baking a pie. When she bakes, she tends to hum songs. When she bakes, I tend to smell it happening. When she bakes, I tend to see her less in the living room. I am sitting in my armchair observing the smell…
Pavel Komarov
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