# 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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### If multiple variables add up to 1, are they independent of each other?

I am trying to test for association between continuous fractions of cell types in a sample (e.g. immune cells, cancer cells, fibroblasts...) and tumour grade (categorical/binary/ordinal, grade 1 or 2)....
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### Can I use a Mann-Whitney U test to compare 2 small groups, with unequal amounts of data, non-normal distributions, and unequal variance?

I want to compare data between 2 independent groups to determine whether there is a significant difference between the groups. 1 group has 8 data points, and the other has 9. Separately, I would like ...
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### Does homocedasticity and conditional mean of error equal 0 imply the error is independent from the explanatory variable?

Let $Y = X \beta + e$ be a regression model, where $E[e|X] = 0$ and $E[e^2|X] = \sigma^{2}$. Does this mean that $e$ is independent of $X$? If so, how to prove it? If not, are there some ...
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### What statistical test of significance should I use when comparing two small, unequal groups of non-normal distribution?

I am trying to analyze quantitative data between two independent groups. One group has 8 data points, and the other has 9. I used a Shapiro-Wilk test for each of the groups to determine normality, and ...
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### Independence of Random Variables - from t distribution and its definition [duplicate]

I have a somewhat strange question about the independence of random variables. It comes from the definition of t-distribution. In this definition, we need two independent random variables and we can ...
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### understanding "independent" term in "independent and identically distributed" (iid) statement

I understand independent as the random variable values in a data aren't connected to each other in any way. (q1) Is my understanding ok? data_1: a product which has a design defect has been sold to ...
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### some thought about independence and orthogonal, please comment on this if it's wrong

It seems that linearly independent is totally different from independent of random variable concept. Non-zero vectors Orthogonality must imply linearly independence. In Statistics, the relation of ...
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### Is this a case of a confounding variable and how can I handle this?

Say I want to predict the speed of an airplane based on the engine power and the weather conditions using a simple linear model. To model the weather conditions, we have measurements of the amount of ...
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### Not necessarily conditionally independent = dependent?

After concluding the d-separation procedure (ancestral graph -> moral graph -> removing directed links), I am left with two nodes that are connected and a conclusion that they are "not ...
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### Independence does not imply Zero Correlation

If I take $X$ to be a degenerate random variable, i.e. $X=1$ WP1 and $Y=X$ defined over the singleton sample space $\Omega=\{1\}$. Then $$\mathbb{P}(X=1|Y=1)=1=\mathbb{P}(X=1)$$ i.e. I'd assume they'...
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### Bagging dependent data

Which are the possible caveats of using a Bagging algorithm (such as Random Forest), when data are not independent? Ensemble models usually exploit Bagging to reduce the variance by aggregating ...
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### Limitations of normal distribution with non independent mean and variance

If $X\sim\mathcal{P}(\lambda)$ and $\lambda\geq 1000$, then one says that $X\approx\mathcal{N}(\lambda, \lambda)$. But, by reading about confidence intervals for normal distributions, the calculation ...
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### Assumptions of compound Poisson model

My understanding of a compound Poisson RV is one defined as $$Y=\sum_{n=1}^N X_n$$ where $\{X_n\}_{n\in\mathbb{N}}$ is a sequence of identically distributed and mutually independent (iid) RVs $N$ is ...
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### From marginal distribution to joint distribution with independence

Consider a random vector $(X,Y,Z)$, Let $f_X, f_Y, f_Z$ be the probability distributions of each component. Question: Does there always exist a distribution $f$ for the whole vector $(X,Y,Z)$ such ...
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### CRLB derivation - Estimator indepedence of estimated param

I have followed the CRLB derivation, and I couldn't figure out why - If f(x; θ) be a probability density with continuous parameter θ, and X1, . . . , Xn be independent random variables with density f(...
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### If X and Y are independent random variables and X is independent of f(X,Y), what can be said about f?

I know that if $X$ and $f(X)$ are independent, then $f$ is almost surely constant. Thomas Lumley has pointed out that very little can be said about $f$ without additional constraints: what if the ...
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### Is the sample independent from the mean?

I came across this question in Dobson's Introduction to Generalized Linear Models: What concerns me is that statement that the $\bar{Y}_j$ and $Y_{jk}$ are independent. How can $\bar{Y}_j$ (the ...
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### Proving Independence due to exchangeability?

I have a set of bernoulli random variables $\{x_i\}^{n}_{i=1}$ and $\{x_{ij}\}_{i< j}$. They have a probability distribution with following conditional independence: P(\{x_i\}^{n}_{i=1},\{x_{ij}\}...
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### What is emperical version of Kendall's tau and can be used as a dependecy measure?

I know Kendall's tau as a commonly used measure of dependency. I read that it is inconsistent as an independence test, and hence, another version of consistent Kendall's tau has been introduced and ...
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### Bootstrap method for chi squared test of independence

I really need some advice about using the chi-squared test of independence. I want to use the bootstrap-chi-squared method for conditional independence testing. The problem is that the DOF is really ...
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### Poisson regression and independence

I'm using Poisson regression to test the relationship between root number and canal number in human teeth. My question is about the independence of variables, particularly – a) does including multiple ...
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### Independence of events in repeated trials

Just a bit confused about whenever "Independent" event is used in probability/stats videos/books. When we say an "Independent" event, are we referring to an event standalone or in ...
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### Proof: If $X_1$ and $X_2$ are independent random variable, are $X_1$ and $X_1X_2$ independent? [closed]
I am working through a set of example problems. I am stuck on this on which defines the distribution are: $P(X_i=-1)=P(X_i=1)=\frac{1}{2}$ for $i=1,2$. I am trying to show whether $X_1$ and $X_1X_2$ ...