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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How to calculate the expectancy of the ratio of non-independent random variables?

How can I calculate this expectancy: $$ E \left [ \frac{\sum_{t=1}^T{Z_tX_t}}{\sum_{t=1}^T{Z_t^2}} \right ] $$ where $Z_t \sim N(0,1)$ and $X_t \sim N(0,1)$ are independent? Any tricks? Is it ...
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U -statistics for bi variate sample problem

Let $(X_1, Y_1), (X_2, Y_2),....,(X_n, Y_n)$ be iid random variables with joint distribution function $F(x, y)$ and $F(x), G(x)$ be the marginal distribution functions of $X_1$ and $Y_1$ respectively. ...
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Does independence almost everywhere imply independence?

Let be $X$ and $Y$ two random variables such that, for any event $A$, $P( X \in A \mid Y) = P(X\in A)$ with probability 1. Can I conclude that $X$ and $Y$ are independent ?
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PMF of the Independent Multivariate Bernoulli Distribution

I was reading this paper on the Multivariate Bernoulli Distribution, which provides the general form of the PMF in equation 3.1. The paper refers to this as the probability distribution function, but ...
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Consequences of maintaining IID assumption for prediction model training, but relaxing it for model testing

Let's say you're developing a prediction model, and you are confident that your data are IID. For example, you have a dataset where each row represents a different patient, and you build a model to ...
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external internal validation and chi square

The internal test set, created from 20 percent of the train data set, consists of 'a' and 'b' labels. I take the 'a' labels from the internal test and combine them with the 'c' group of another ...
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Undirected graphs and implications of independence (Wasserman chapter 18)

In Wasserman's All of Statistics chapter 18, he defines the following undirected graph: Let $V$ be a set of random variables with distribution $\mathbb{P}$. Construct a graph with one vertex for each ...
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How to find the variance for $\frac{\sum Y}{\sum X}$ when X and Y are both independent and normal random variables

The original problem is: Given $Y_i = \beta X_i + \epsilon_i$, $i=1,2,...,n$, where $X \sim N(\mu, \tau^2)$ iid and $\epsilon \sim N(0, \sigma^2)$ iid, $X$ and $\epsilon$ are independent. What is the ...
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DEPENDENCE AND CORRELATION. biostatistic italian test [closed]

In the test there are these 2 questions. My friends and i cannot tell the correct answer because it is so confusing. It is a multiple choice test. Are two dependent variables correlated? A. always B. ...
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How to show in R, using a simulation, that when sampling from a normal distribution, the sample mean and sample variance are independent

What is the best way to show that when sampling from a normal distribution, the sample mean and sample variance are independent? I know the theory behind this result, I would like to show it using a ...
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Join distribution of independent random variables that aren't conditionally independent

I am asked to give an example for a joint distribution of three random variables, $U$, $V$ and $W$, where $U$ and $V$ are (unconditionally) independent but are NOT conditionally independent given $W$. ...
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Statistical test for small non-parametric dataset with more than 2 dependent groups

I’m trying to figure out the most appropriate test to use for a small water quality dataset (n = 10 sampling visits at 6 river sites, upstream to downstream) with the following characteristics: -not ...
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Clarification on covariance matrix for multidimensional Gaussian distributions

It is a well known property of Gaussian distributions that if $Y = (Y_1, \ldots, Y_n)$, where each $Y_i$ is a real Gaussian random variable, then the components of $Y$ are independent if and only if ...
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A and B are independent. Does P(A ∩ B|C) = P(A|C) · P(B|C) hold?

Let $C$, $B$, and $A$ be events in the same probability space, such that $A$ and $B$ are independent and $P(A \cap C) > 0$, $P(B \cap C) > 0$. Prove or disprove: $P(A \cap B|C) = P(A|C)P(B|C).$
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ANOVA with variables with known (but arbitrary) conditional dependencies

I have a dataset with the following properties: k > 2 groups normally distributed differing variance and sample size between groups non-independent samples within each group continuous variable ...
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Independence of 2D gaussian process derivatives

Suppose I have a gaussian process which takes 2D inputs x and y and gives a 1D output z. I understand based on Calculating the expression for the derivative of a Gaussian process that each of the ...
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If $A^2$, and $B^2$ are DEPENDENT random variables, will $A$, and $B$ be necessarily DEPENDENT too?

I know that if $A$, and $B$ are independent, the independence is preserved for $A^c$, and $B^c$, where $c$ is a constant. I am wondering if the same applies to the case where the random variables are ...
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Correlations among rankings

In my study, participants ranked their preferences for several choices. Because moving one choice up necessarily means moving one or more choices down, nearly all correlations (Pearson's r) are ...
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Can a Dependent Sample T-Test be Used on Sample Group (A) of 100 Electrical Devices, Tested at time, t1, shuffled, and Tested at time, t1?

Description: I have a group of 100 electrical parts being testing for Forward Voltage, at time, t_1. This is my sample group, S1. This same group is undergoing a stress test that may or may not affect ...
randomguyz's user avatar
2 votes
1 answer
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Chi square test (or alternative) for mixed design categorical data

I have the following question: I collected the number of different symptoms at two time points (Baseline BL, Follow-up FU) in two groups (Control Group CG, Intervention Group IG). So, there is a ...
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Independence across observations [duplicate]

For various reasons we often assume that there is independence across observations in linear regression models. When does that assumption hold? Only when the data is collected by random sampling? I ...
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Probability that X > Y when X ~ N(0,2) and Y ~ N(0,1)

$X$ and $Y$ are independent variables $X$ ~ $N(0,2)$ and $Y$ ~ $N(0,1)$. What is the probability that $X > Y$? I understand that the distribution of $X - Y$ ~ $N(0,3)$ assuming they are independent....
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Markov Chain and deterministic function

Here is a problem I am trying to solve: Consider a sequence of IID random variables $Y_1,Y_2,Y_3,...$ with values in $E$ and let the function $\varphi: E^2 \rightarrow E$ define the corresponding ...
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Are mutually exclusive events are independent events? [closed]

So far my understanding: consider two mutually exclusive events, A and B. If A occurs, B cannot, and vice versa. In this case, the occurrence of one event (say, A) provides certain information about ...
Zahid Hasan's user avatar
1 vote
1 answer
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Wide to long data - sort of [closed]

I have data that essentially looks like the following: Patient X1 X2 X3 101 -3 -1 2 102 -5 2 -1 103 1 3 2 ,,, I can normalize X1, X2, X3 so ...
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Alternative to paired t-test for non independent data

I'm working on a research project where I predict fMRI activity using two separate sets of predictors, X1 and X2. In order to do this, I am fitting a linear regression from each predictor to each ...
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Is there a known way of producing forecasts with reasonable fit and residuals that are at least independent, & ideally negatively correlated? [closed]

I am trying to do some forecasts. I have produced multiple forecasts by a variety of methods. All of the forecasts I have generated so far have residuals that are strongly positively correlated.I ...
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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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Basic question regarding before and after intervention analysis on groups of different sample size

Apologies for the basic nature of this question. I have collected data before and after an intervention and assessed response to the intervention using a survey. This was a basic survey using a 1-5 ...
june2023's user avatar
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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,...
Dylan Alves's user avatar
2 votes
1 answer
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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 $P(X>k)*P(Y>z)=P(X>k,Y>z)$? [duplicate]

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 ...
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Target variable is defined by combination of input features

I am trying to create a classification model which predicts whether or not a customer comes back to make a second transaction (after having made an initial transaction). I have details on date of ...
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How to interpret multinomial logistic regression results in r with ordered independent variable?

If there is a independent variable X that is a ordered factor (low, medium, high) - high being the base class, the results of the multinomial regression for each class in X should be interpreted as ...
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Accounting for occasionally correlated data in performance analysis

I'm doing a study in which a new diagnostic method is evaluated. Samples are taken from patients, both this new method and the current standard method are applied, and we calculate concordance between ...
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how to account for non-independence in the analysis of dissimilarity matrices with restricted PERMANOVA or alternative methods

I am doing a study looking at how bacterial communities on eggshells correlate with species-site and nest stage. We sampled the eggshell bacteria at two sites, with two different host species sampled ...
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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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p-value of pearson correlation of correlated samples

I'm not completely sure if this question has been asked before, but I couldn't find one that suit my problem so far. So basically I am doing single cell analysis, and I have roughly > 100000 cells ...
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Joint probability based on assuming linear relationships

I am a statistical novice trying to work something out which is probably basic. Or impossible. I have discrete probability distributions for two random discrete variables $A$ and $B$. I want to find ...
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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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