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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Confusion about assumptions in classification problems
I was studying Linear Discriminant Analysis, and this general case came up which used Bayes theorem.
Suppose we observed response values of $Y \in \{0,1\}$ and predictors $X \in \mathbb{R}$. Suppose ...
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Can a t-test be used to compare means obtained from samples of the same batch but tested in two different labs? [closed]
I have two datasets containing size measurements. The initial dataset originates from six different aliquots drawn from a large batch of a product (a suspension with particles). Each sample was ...
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What test should be used for two non-independent groups of different N?
This is my first question so apologies for any problems with it.
I am carrying out an experiment which involved counting the number of people using the stairs or the elevator during a 30 minute period,...
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Source for Bernstein's example for pairwise independence
The German Wikipedia gives the following example for events that are all pairwise independent, but not jointly independent:
From four paper slips containing the numbers 112, 121, 211, and 222, one ...
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Two events, both of probability zero, have caused an outcome of probability zero. Which of them did happen?
Let $X$ and $Y$ both be standard normal distributions - so with mean 0 and variance 1, and independent of each other.
Now let $Z = XY$.
We know that $P(Z=0) = P(XY=0) = 0$, because the set $\{0\}$ is ...
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How to consider independent samples in a mixed-design experiment?
I collected data from an experiment with a mixed-design, specifically a 2x3x4 mixed-design, where the factors are populations (2 levels), conditions for the first population (3 levels), and conditions ...
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Why is $P(A \mid C \cap B) = P(A \mid C)$ true in this instance?
As I was reading through this paper http://www.jstor.org/stable/25652278 I came across the following problem:
Consider an urn with $N$ colored balls, the number of red balls, $X$, has a binomial ...
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Given any events $A$,$B$ and $C$ where $A$ and $B$ are independent, is it true that $P(A|B\cap C) = P(A|C)$?
It seems to make intuitive sense that if the events $A$ and $B$ are independent then $P(A|B\cap C)=P(A|C)$ because the occurrence of event $B$ should not change the probability of event $A$ even when ...
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If random variables X,Y are independent is $P(X>k)*P(Y>z)=P(X>k,Y>z)$?
If random variables X,Y are independent is $P(X>k)*P(Y>z)=P(X>k,Y>z)$?
I know if X and Y are independent then $P(X=k)*P(Y=z)=P(X=k,Y=z)$
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If $X_1, \dots, X_n$ iid, are $f(X_1), \dots, f(X_n)$, also iid? [duplicate]
If I have independent and identically distributed random variables $X_1, \dots, X_n$, then are $f(X_1), \dots, f(X_n)$ themselves independent and identically distributed?
I think the answer is yes, ...
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Measuring independence
I want to determine whether my assumption that the dataset I'm using is i.i.d. is in fact valid (for an arbitrary dataset, perhaps made of images).
I have done quite a bit of research already, looked ...
PackDude
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Lower bound on probability that pairwise independent Bernoulli random variables sum to 1
I'm trying to find a lower bound on the probability that $k$ pairwise independent Bernoulli random variables with $p=\frac{1}{k}$ sum to 1. The probability that they sum to $>1$ is upper bounded by ...
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How should I test whether simultaneous residuals from two models of the same time series are independent?
Suppose I have two different models, with comparable goodness of fit but very different structure, that I have fit to the same time series. For both, the residuals pass various tests of normality. How ...
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Dependence or independence of three random variables
Consider I have three random variables A, B, C. I know that A depends on (B,C). Can I always deduce that it implies that A depends on B and also A depends on C? I mean does it implies that neither A ...
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The third central moment of a sum of two independent random variables
Is it true that in probability theory the third central moment of a sum of two independent random variables is equal to the sum of the third central moments of the two separate variables?
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Conditional expectation and independence
Consider 3 random variables $X$, $Y$, $Z$. Suppose:
$E(X)=0$
$E(X|Y) =0$
$Z\perp Y $
Does this imply $E(X|Y,Z) = E(X|Z)$?
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Are my data independent?
If I wanted to analyze data taken of patients meeting pre-specified criteria from several hospitals through a linear regression (say how hypertension changes given age and sex), I think that a mixed/...
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Conditional distribution $f(x|y)$ if $X$ and $Y$ are independent
Suppose we have two randome variables $X$ and $Y$, with joint distribution $f(x,y)$. $X$ and $Y$ are independent if and only if the marginal distribution of $X$ is the same as the conditional ...
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Effective sample size after applyting butterworth filter
I have a signal s with a total length of N*D in time, which I break into N observations of ...
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All uncorrelated marginals are independent: Only for joint Gaussian?
Let $X$ be a random vector in $\mathbb{R}^p$, where $p\geq 2$, with the following property: Any two uncorrelated marginals are independent. Formally:
(1) For any $\alpha,\beta\in \mathbb{R}^p$, if $...
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Violation of i.i.d assumption of supervised learning models with time series data
I am trying to develop a model that predicts the number of cable and joint faults in a distribution grid on a daily basis. These faults seem to increase during very hot and arid summer days (Heatwaves)...
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Different amounts of observations for individuals - Which model to use?
I have a dataframe which is structured in a way, that it includes a different amount of observations (trip chains) for each person. A trip chain consists of all the trips an individual took from ...
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Analyzing multiple data points per condition of a simulation model
I wish to perform statistical analysis on a data set that is the result of running a simulation model.
I have simulated this model by varying five parameters, TA_PS, TA_IP, PA_PS, PA_IP, and PA_LGPP. ...
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Is the pairwise independence gap bounded to $\left[-\frac{1}{4},\frac{1}{4}\right]$? What about for n variables?
The independence gap is defined as
$$\phi_{X_1, \ldots, X_n}(x_1, \ldots, x_n) \triangleq F_{X_1, \ldots, X_n}(x_1, \ldots, x_n) - \prod_{j=1}^n F_{X_j}(x_j)$$
where $F_{X_1, \ldots, X_n}(x_1, \ldots, ...
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Is there any measure that tells you if two variables are dependent when there is no linear or monotonic relationship?
We know that Pearson correlation tells you the linear correlation (proportionality) between two variables, meanwhile Spearman rank correlation tells you how far you have a monotonic relation (if one ...
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Does it make sense to compare two different DVs on a within-subjects t-test or MANOVA (or ANOVA?)?
Let's say I have 3 different drug doses: A, B and C and I have two DVs: blood pressure and pulse. I am interested in finding out if the three doses differentially impact the two DVs (i.e. if at dose A ...
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Independence and conditional independence in probability
In a book, to explain independence and conditional independence in probability, the author gave an example that, consider the following situation:
F={ a hospital patient is smoker}
G={ a hospital ...
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A question about independence: if $X \perp Y$ and $X_1 \perp Y_1$, then $XX_1 \perp Y Y_1$ [duplicate]
Let $X$ and $Y$ be independent random vectors (denote this by $X \perp Y$ ) . Let $A=[|X|\leq t ]$ and $B=[|Y|\leq t ]$. Since
$X\perp Y$, we know that $A$ and $B$ are independent, so the indicators $\...
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Puzzling negative correlation
Question edited with the correct code - apologies
I have observed a somewhat puzzling negative correlation.
In code (R) (mathematical formulation below)
...
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Testing for conditional independence with nonlinear relationships
I am reading about the IC and IC* (Inductive Causation) algorithms for discovering DAGs from observations. The first step of the algorithm is for each pair of variables a and b, search for a set of ...
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Given a DAG X<-Z->Y, how to test for conditional independence of X and Y given Z for non-normal random variables?
Consider data generated from a directed acyclic graph (DAG) of the form X<-Z->Y. Based on d-separation of DAGs, X is independent of Y given Z: X_||_Y|Z. If X, Y and Z are multivariate normal, ...
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Are paired matches in ranked data independent, if the original, unranked, continuous pairs are independent?
Let $(X, Y)$ be a randomly drawn sample of $n$ paired observations from a bivariate continuous population. It is clear that each pair is independent of the others, both pairwise and mutually. Let $R(X,...
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Are moduli of components of Fourier transformed Gaussian random vector still independent?
Suppose $X=[X_1,\ldots, X_n]^T$ is a random (column) vector such that:
$X_i \stackrel{i.i.d}{\sim} \mathcal{N}(0,\sigma^2), \ 1 \le i \le n$
$\mathcal{F} \in \mathbb{C}^{n \times n}$ is the discrete ...
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Definition of events in independence simple coin toss
I struggle defining the experiment, the sample space and the events.
For instance, I have two ways of defining the following:
I have the experiment of tossing a coin twice, my sample space would be {...
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Conditional expectation of a linear combination of random variables given one of the random variables
I recently got confused by an expression I found in a paper,
$$E[aX + bY \mid Y] = aE[X\mid Y] + bY, $$
I am confused that it's true if $X$ and $Y$ not independent. What would be the decomposition if ...
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Representation of two Gaussian vectors as sums of independent Gaussian vectors
I know that if we have 1-dimensional Gaussian r.v. X and Y we can find coefficients $a,b$ so that $$X=aY+bZ+E[X-Y]$$ where $Y , Z$ are independent and $Z$ is standard Gaussian.
Can we do something ...
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Test for independence of trends
Suppose I have two distinct outcomes being tracked over time, A and B. An individual observation can have outcome A, outcome B, both (call it AB), or neither.
If the rate of outcomes A and B are both ...
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How can I introduce dependence (to varying degrees) into a synthetic dataset to measure the effect on my method?
I'm using a synthetic dataset in which I sample from three independent Bernoulli random variables x1, x2 and x3 with p=p1, p=p2 and p=p3 respectively. I wish to "introduce dependence," or ...
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Evaluating linear regression model [duplicate]
I understand that when performing linear regression, one common rule-of-thumb is that for a good 'fit', the residuals should be 1) independently distributed, 2) stationary and 3) not serially ...
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Conditional independence statements for probabilistic motivation for linear regression
So the motivation for using the squared loss in linear regression can be written as the following (I think):
Assume $\{(\mathbf{x}_i, y_i) \mid i = 1, \dots n\}$ are repeated independent samples from ...
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How can principal components be correlated when averaged?
I am working with sequences that consist of 20 building blocks. Those 20 building blocks were characterized using different parameters and after linear PCA, the first 3 principal components were ...
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meaning of independent samples
I am getting confused about the concept of independent samples, which is related to the concept of pseudoreplications. This question is related to this topic. But reading it does not make it clear for ...
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Is it valid to test the divergence of two timeseries using a Mann-Whitney-U test? If not, how can I show doing so will be problematic?
I have two time series that give me a monthly count. One is a reference series that is meant to be used as a control; the other has a 'treatment' or program applied to it mid-way through. Both are ...
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Shapley values for groups of correlated features
Is there a version of Shapley values that does not assume independence of features, and can be used to interpret importance of clusters of "similar" features (by adding individual feature ...
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Does statistical association and dependence only include linear and nonlinear correlations
I know there are linear correlation measures such as Pearson that might not capture all cases of statistical dependence or nonlinear correlations. I know there are also nonlinear correlations, which ...
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Prove that two random variables are independent
I am struggling with the following problem (casella & berger 4.30(b)):
$$ \text{Suppose that} \;\;\;Y|X=x \sim normal(x,x^2) \;\; \text{and} \;\; X\sim uniform\,(0,1).\\\text{Prove that} \;\;\frac{...
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Calculate joint distribution from marginal distributions
I am struggling with the following problem: $X_1, X_2 \sim N(0, 1)$ are independent random variables. Let $Y_1 = \frac{1}{\sqrt{2}}(X_1 + X_2)$ and $Y_2 = \frac{1}{\sqrt{2}}(X_1 - X_2)$. Show that $...
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Prove that white noise + normality = independence
If the time series process is linear, then the ARIMA model is specified.
The residuals from this model are $(1.)$ no autocorrelation $(2.)$ mean equals zero $(3.)$ constant variance. We say that this ...
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Are linear combinations of independent random variables again independent?
Let $X_1,X_2,\ldots,X_n$ be (iid) Random variables and define $Y_n:=\sum_{j=1}^na_jX_j$ with $a_j\in \mathbb{R}$, can we then say that the $a_jX_j$ are independent aswell. Can we express the MGF than ...
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Let $X_1,X_2,\ldots$ be iid random variables with Cauchy distribution and $S_n=X_1+X_2+\cdots+X_n$, find $P(S_n>an)$, $a>0$
Let $X_1,X_2,....$ be iid random variables with Cauchy distribution and $S_n=X_1+X_2+\cdots+X_n$, find $P(S_n>an)$, $a>0$. This is exercise 8.44 of the intro book of Grimmet and Welsh. We cannot ...
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