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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Causal discovery for pairwise independent joint dependent variables

Consider the standard example for variables that are pairwise independent but joint dependent. $$ (x,y,z)= \begin{cases} (0,0,0) & \text{probability 1/4} \\ (1,1,0) & \text{probability 1/4} \\ ...
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Are these data pseudoreplicated?

I have an amalgamation of independently collected datasets covering a large area of ocean where surveys to count birds have occurred sporadically over a long time. Some surveys were from boat, some ...
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Efficient ways to measure the degree of independence of a moderately large number of variables

I have a process that generates values for variables $x_{1}, x_{2}, \dotsc, x_{n}$ where $n \approx 40$, and the value of each $x_{i}$ lies between $0$ and $1$. The process generates these in batches ...
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Suggestions for an independence test in a complex design

In an experiment I surveyed the effect of two treatments (pre & post) in different species. After every experimental run I tested whether the measured average effect was greater, smaller or not ...
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Show that no two sets in the probability space with $\mathbb{P}(\{k\})=2^{-k!}$ are independent

Let $\mathcal{P}(\mathbb{N})$ denote the power set of $\mathbb{N}$. Show that no two non-trivial sets in the probability space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mathbb{P})$ with $\mathbb{P}(\{k\})=...
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How to analyse serial brain sections probed for different proteins (ttests and potenial issue of independance)

Generic scenario: Brains have been collected from two different populations. They then have been cut into thin sections (serially) for the purpose of looking at the expression of proteins in specific ...
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Studying extreme value r.v. $X=\max_i (c_i+X_i)$ where $c_i$ are constants and $X_i$ are i.i.d. r.v

Let $X_1,X_2,...,X_n$ be independently and identically distributed random variables according to a distribution $F$. There are constants: $c_1,c_2,...,c_n$. Define a new random variable $X=\max_i(X_i+...
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"Predictive dependence" between two variables

Given two random variables $X$ and $Y$, it is natural to use the conditional entropy $H[Y|X]$ to quantify the extent to which knowing $X$ decreases the uncertainty about $Y$. However, consider the ...
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P(XY<1/2) , X and Y uniform independent RV [-1,1]

I think the probability should be equal to shaded area shown in the figure divided by 4, which is equal to (3+ln 2)/4. Is this correct, because my answer is not matching My solution: I find the area ...
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BayesNet Independence

For BayesNet, can anyone explain how we can check the independence between the set of random variables? e.g. $\{B, D\} \perp \{G, I\} | A?$
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P-value of chi-square test of independence

I need help understanding chi2 independence test scipy.stats.chi2_contingency. Let's assume I have two samples (of different sizes) of a categorical variable with 3 possible outcomes (1, 2, 3): Counts ...
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ANOVA vs Chi Square Test to compare one independent and one quantitative variable

I believe I understand the basics of chi square test and ANOVA but not confident enough to make a decision on which test to apply or whether both tests can be used. I have a categorical variable: ...
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Are $U=\frac{2X_1^2}{(X_2+X_3)^2}$ and $V=\frac{2(X_2-X_3)^2}{2X_1^2+(X_2+X_3)^2}$ independent?

Consider i.i.d standard normal variables $X_1,X_2,X_3$. How can I determine whether $U=\frac{2X_1^2}{(X_2+X_3)^2}$ and $V=\frac{2(X_2-X_3)^2}{2X_1^2+(X_2+X_3)^2}$ are independently distributed? This ...
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Fitting the most appropriate model for the data

To compare 4 detection machines and also the performance of 3 scientists, the amount of gold within a chemical substance (of fixed weight) was determined by each scientist using each machine. Two ...
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Joint distribution of sample correlations of variables taken from a multivariate normal distribution

Let us assume multivariate normal vector $(X_1, \cdots, X_n)$ with mean vector $\mu$ and variance-covariance matrix $\Sigma$. A sample correlation will not exactly equal its population parameter, but ...
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Does the marginal distribution assume independence? (i.e. sampling without replacement)

Let $X_1, X_2, ... , X_n$ be a sample drawn without replacement from a finite population. $X_1$ may be the random variable - weight of the first person; $X_2$ may be the random variable - weight of ...
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What is the purpose to have the "independent" covariance structure in GEE or GLS?

The methods of estimation like GLS or GEE are especially helpful, when there are clusters of data, like repeated observations, many per cluster=subject. Such observations are naturally correlated in ...
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Does zero cumulants imply independence?

Question: Suppose we have two random variable $X$, $Y$ that follow non-Gaussian distribution, and we are given that: $$\operatorname { cum }(X, Y)=\operatorname { cum }(X, X, Y)=\operatorname { cum }(...
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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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1 answer
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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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2 votes
1 answer
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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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Does non-zero Spearman rank correlation imply dependence of original variables?

Introduction Let say we have random variables $X$ and $Y$, and we take their rank transforms to be $g(X)$ and $g(Y)$. The Spearman rank correlation coeffiicient can be considered to be $$R[g(X),g(Y)] =...
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Testing whether variables are identically and independently distributed

I have the probabilities of variables that we assume initially are independently and identically distributed random variables. I know that in actuality they are not from my knowledge of the problem, ...
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A question about conditional expectation involving independence

If the vector $(u,v)$ is independent of the vector $x$, then I would like to show that $$E(u|x,v)= E(u|v)$$ The only thing I can derive from the definitions is that if $(u,v)$ is independent of $x$, ...
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How to guarantee the test set is "independent"?

In Machine Learning (ML) tasks, one splits the dataset into training and test sets. We train the ML model based on the training test, and then we evaluate the performance of the model with the test ...
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Arithmetic on Equalities in Distribution: Coincidence?

I gather that if we have random variables such that $X_1,X_2$ are independent and $Y_1,Y_2$ are independent, then $$X_1=_dY_1,X_2=_dY_2\implies X_1+X_2=_dY_1+Y_2,\quad (1)$$ where $=_d$ denotes equal ...
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Are my variables dependent or independent?

I have 2 sets of variables that I need to run tests on and I am not sure if they are dependent or independent. I will describe them below: Set 1: these values are river water samples that I collected ...
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2 answers
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Regression (in a wide sense), homoskedasticity and independence of error term

I suspect this might have a simple answer, but I have been stuck on it for a while. It is a simple true or false question. I suspect it to be false, but I haven't come up with a counterexample. At ...
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1 vote
1 answer
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Proof that each random variable $Y_{j}$ has the same probability distribution

Let $X_{1},X_{2},\cdots,X_{n}$ be nonnegative, independent and identically distributed random variables.Show that,if $k\leq n,$ then $$\mathbb{E}\left(\frac{X_1+\cdots+X_k}{X_1+\cdots+X_n}\right)=\...
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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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2 votes
1 answer
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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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1 vote
1 answer
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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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2 votes
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Independence of a random variable jargon

Just a bit confused about whenever "Independent" random variable is used in probability/stats videos/books. When we say an "Independent" random variable, are we referring to a ...
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3 votes
2 answers
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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 $X_1X_2$ ...
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