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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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Degree of dependence between individuals of a variable

I'd like to know if there is a measure that computes the dependence between individuals of an only variable. For example, let $X$ be such that $X = \{x_1, \ldots, x_n\}$, I'd like $dependence(x_1, \...
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Gaussian distribution: moments, independence and rotation

I have a few questions with respect to the gaussian distribution, its moments and independence. So a gaussian distribution is fully specified by its first two moments, the mean and variance (or ...
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106 views

Mutual Independence in a Multivariate Normal with Identity Covariance

Consider a random vector $X$ which follows a multivariate nomal with zero means and Identity Covariance. $X\sim \mathcal{N}_n(\mathbf 0, \mathbf I)$ We can say that the individual variables $X_1, ...
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Does the independence of samples in an experiment matter?

This is a bit complicated. I have a set of data of events happening in time, so my data looks like an array of time points at which an event happened. (Eg [2 45 50 51 60 79] in seconds) This data was ...
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Statistical test for independence of a sequence of behaviours

I collected data from a social experiment. The data consists of a set of interaction behaviours of the participants. Each participant can perform 2 behaviours, either A and B, and they can perform ...
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17 views

Dependant or independant observations

I am reading a study where an area of forest is subdivided into different plots and the plots are randomly assigned treatments. I am wondering whether the fact that the plots all come from the same ...
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Proving that $ (\hat{\beta} - \beta)' (X' X) (\hat{\beta} - \beta)$ is independent with SSE

Exercise: Prove that $ \mathbf{(\hat{\beta} - \beta)' (X' X) (\hat{\beta} - \beta)}$ and SSE are independent for a Least Squares Regression Model. Attempt: Note that by $'$ I denote the transpose ...
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How to derive $\operatorname{var}[(X_i−\mu)^2]=2\sigma^4$ where $X$ is distributed normally

I have $X_1,...,X_n$, i.i.d. $N(\mu,\sigma^2)$ and I would like to calculate $\text{var}[(X_i−\mu)^2]$. I know that the solution is $2\sigma^4$. However, I can't derive it. Any suggestions?
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2answers
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Prove that the sum and the absolute difference of 2 Bernoulli(0.5) random variables are not independent

Let $X$ and $Y$ be independent $Bernoulli(0.5)$ random variables. Let $W = X + Y$ and $T = |X - Y|$. Show that $W$ and $T$ are not independent. I know that I have to show that $P(W, T)$ is not equal ...
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18 views

Conditional independence of two events

1) If events A and B are independent on given condition C, then does it implies that those two events A and B are independent without the condition C? 2) If events A and B are independent events, ...
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Statistical test to compare dependent samples

I am comparing by race the number of limbs (4 total - including both arms and both legs) affected by condition X. Some patients contribute 4, some 1, and some somewhere in between. I want to compare ...
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To find the covariance given the joint probability density function.

Question: I was solving some question papers and got stuck in this problem. My problem: I know how to find "marginal probabilities" from a joint probability density function and also know how to ...
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2answers
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Does independence between random variables imply independence between related events?

Say I have two random variables X1 and X2 and that they are independent. Am I guaranteed that the events "X1 is less than x1" and "X2 is less than x2" are independent? If not, under which conditions ...
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1answer
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Association rule lift ratio converging to 0.5, is it not independent?

As far as I know, lift is a measure used in association rules to see whether there is a positive association between two items(instances?) I've done a simple (possibly wrong) simulation with lift ...
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Infinitesimal independence

Let's say we have two random variables $X$ and $Y$. Is there a name for saying that $X$ and $Y$ are independent only for the values concentrated around a small interval around some $x_0$ and $y_0$ ? ...
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1answer
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Survival model for an epidemic — can the observations be treated as independent?

I've been thinking about ways to tackle an epidemic modelling problem I've been working on, and I've come up against a conceptual difficulty over the way survival analysis works. Here's a really ...
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Imposing independence constraints in mixture modeling of correlated data?

For 1-D signals (spectra) or 2-D signals (images), is there a way to impose the constraint that the data within a group is uncorrelated? I am iteratively applying background correction model fitted to ...
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synthetic datasets without any dependecies between features

I have to create a synthetic data set without any dependencies between features so that this equation should be hold. I thought about to take simply several Gaussians or randomisers, each would be ...
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67 views

Expected value of quotient of Poisson distributions

Let $X$ and $Y$ be independent random variables such that $X \sim \text{Poisson}(\lambda \cdot c)$ and $Y \sim \text{Poisson}(\lambda \cdot (1-c))$, where $c$ is a real number in $[0, 1]$. Is there ...
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Are my two samples independent enough for a Kolmogorov-Smirnov test?

I have a set of polygons that correspond to home ranges of, say, 100 individuals for which I have measured the overlap between all possible pairs. This has given me a square matrix of dimensions ...
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241 views

Uncorrelatedness + Joint Normality = Independence. Why? Intuition and mechanics

Two variables that are uncorrelated are not necessarily independent, as is simply exemplified by the fact that $X$ and $X^2$ are uncorrelated but not independent. However, two variables that are ...
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Prove the relation between two distribution functions

I have been given a homework in a subject called "Non-Parametric Statistics" and I'm a bit stuck with it. I would be very thankful if you could give me any advice or help, which would lead to a ...
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Weak independence of gaussian distribution given a constraint on samples

Suppose samples $x_1,x_2,\ldots,x_n$ in space $G$ follow a (multivariate) gaussian (normal) distribution with specified mean and variance, i.e., $x_1,x_2,\ldots,x_n \overset{i.i.d}{\sim} f_X\sim N(\...
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2answers
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Definition of independence of two random vectors and how to show it in the jointly normal case

(1) What is the definition of independence between two random vectors $\mathbf X$ and $\mathbf Y$? I think it's more than just pairwise independence between all the elements $X_i$ and $Y_j$. (2) The ...
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1answer
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Upper bound on $P(n^{-1}\sum_{i=1}^n (X_i - \lambda_i)>t)$ for independent $X_i\sim\operatorname{Poisson}(\lambda_i)$

Let $X_1,\dots,X_n$ be independent random variables, $X_i \sim \operatorname{Poisson}(\lambda_i),$ $i=1,\dots,n.$ Let $$S=n^{-1}\sum_{i=1}^n X_i, \quad\quad \lambda=n^{-1}\sum_{i=1}^n \lambda_i.$$ ...
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1answer
30 views

Conditional probability and independence

Suppose A and Y are discrete dichotomous variables $(A=0,1; Y=0,1)$ If $Pr[Y=1|A=1] = Pr[Y=1|A=0]$, why can we conclude that $$Pr[Y=1|A=1] = Pr[Y=1|A=0] = Pr[Y = 1],$$ without knowing beforehand ...
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548 views

Can anyone help explain this basic example of posterior

I am having trouble understanding the authors reasoning here. It is from "The Bayesian Choice" I am confused about why the posterior is initially written without depending on the data, and why we ...
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1answer
38 views

Independence of sample variance and $\sum_{i=1}^n w_i X_i$

I have the following model, $$ X_i \overset{iid}{\sim} \mathrm{Normal}(0,1), i=1, \dots, n. $$ It is known that the sample variance $\hat{\sigma^2} := \sum_{i=1}^n \frac{(X-\bar{X})^2}{n-1}$ is ...
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16 views

Independence of random variables. vector z with (u,v)

If one writes that a vector z (elements are random variables) is independent with (u,v) where u and v are random variables, does this mean that z is independent with u, and that z is independent with ...
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1answer
38 views

Showing indepedence of two random variables when $p(x,y) = p(x) \cdot p(y)$ except a constant factor?

During a course I attend at university, I encountered the following question: Given is a probability distribution: $$p(x,y) = \lambda \eta \cdot \exp(-\lambda x - \eta y) $$ supported on $\mathbb{R}...
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Independent and Dependent Random Variables

Please give an example of 2 dependent random variables, X and Y such that P(X < Y)=1. Again, provide an example of 2 independent random variables, X and Y such that P(X < Y)=1 For the first ...
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1answer
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Is the assumption of indepence only for the sampled values informing the regression, or should it also apply to the cells of a prediction grid?

I have 200 discrete, well-spaced plots with reasonably independent sampled values from which I've derived a regression equation. If I use it to predict values on a similarly sized fishnet grid, how ...
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Does a data-dependent sampling rule induce correlation?

[This question is cross-posted on math SE here ] Suppose I have two iid streams of data that are independent of each other: $X = (X_1, X_2, \ldots)$ and $Y = (Y_1, Y_2, \ldots)$. I want to estimate ...
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158 views

Independence of $X+Y$ and $X-Y$

In a roll of die, if $X$ is the number on the first die and $Y$ is the number on second die, then determine whether the random variable $X+Y$ and $X-Y$ are independent. The covariance between the ...
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Compatibility of conditional and marginal independence assumptions

I want to know if two independence assumptions, as illustrated below, would go together or not. Consider I have 4 variables, A,B,C,D. Can the following two independence assumptions co-exist? $A \...
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Can i assume independence after adjusting for cross correlation in an event study?

I'm doing an event study on rebalancing date of an index. The rebalancing happens every quarter. Using 50 tickers this implies i have multiple events on the same ticker, which is fine, but since it ...
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Stuck on a term in $\operatorname{Var}\left[ \widehat{\beta}_0 \right]$ proof

So I was trying to prove that $\operatorname{Var}[\hat{\beta}_0] = \dfrac{\sigma^2n^{-1} \sum{(x_i)^2}}{\sum{(x_i-\bar{x}})^2}$ And I got stuck with the part $\dfrac{-2\bar{x}}{\sum{(x_i-\bar{x})^2}} ...
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$X$ independent of $Y$ conditional on $Y$ in some subset of the domain?

Let $X,Y,\epsilon:\Omega\to \mathbb R$ be random variables. Let's say that $X=\text{sign} (Y) +\epsilon$. Then $X$ is not independent of $Y$. However, we have all the information about $Y$ that we ...
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1answer
37 views

Violation of independence or still a valid test?

I have to test the the effect of two different treatments on a set of samples. On the surface, this seems an obvious option for a two sample test. The problem is that the samples being treated are ...
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1answer
32 views

Sampling from specific distribution

Suppose I have two random variables $X$ and $Y$ that are independent. Also suppose that I can sample from $X+Y$ and $Y$. Is it possible to combine those two sampling algorithms to get samples for $X$.
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1answer
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multivariate Student's t distribution: intuition for non-independence?

Consider a multivariate Student's t distribution, with parameters $\nu$ (d.f.), $\mu$ (location) and $\Sigma$ (shape). Does anyone have a good intuition for the individual components not being ...
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Non-independence of bootstrap samples

Background Suppose we have a dataset that consists of $n$ iid random variables represented as $X_j$, where $j \in \{1,\ldots,n\}$. We know $\forall i,\, \operatorname{E}\left(X_i\right) = \mu$, and ...
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Survival Statistics Independence

If we observe follow-up times Ti = min(Xi,Ci) and right censoring indicators δi = I(Xi ≤ Ci) under noninformative censoring. Is Xi independent of δi? Please explain why, thanks!
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Difference between independence and stationarity tests in time series

This is meant to be a general question, aiming to clarify the topic for a beginner in TSA, as I haven't found any clear introductory explaination yet. Suppose I am working with some data which ...
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3answers
64 views

How to determine if two categorical variables are dependent while controlling for a 3rd categorical?

I have 3 categorical variables: country, gender, and liked (whether the user liked the content or not). Using Chi-squared I see that 'liked' is dependent on country, that 'liked' is dependent on ...
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1answer
35 views

sampling distribution using normal and rectangular distribution

Let $X_1,X_2$ be i.i.d. $N(0,1)$ and $U_1,U_2$ be i.i.d. $U(0,1)$ and independent of $X_1,X_2$. Define $$Z_1=\frac{(X_{1}U_{1}+X_{2}U_{2})}{\sqrt{U_{1}^2+U_{2}^2}}.$$ Find the distribution of $Z$. ...
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1answer
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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 ...
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Questions about tail dependence of copula and copula parameters?

I would like to understand tail dependence and its relationship to the copula function. The relationship between copula and tail dependence can be expressed as: (from this question Understanding ...
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32 views

A symmetric iid process

Let $X_1, X_2, \ldots$ be an iid process with $X_i$ having a symmetric distribution around $0$. Then can I always write $$X_1 - \alpha X_{t-1}-\alpha^2 X_{t-2}-\cdots \stackrel{iid}{=} X_1 + |\alpha| ...
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Showing that two random variables are independent [closed]

If $f(x,y) = g(x+y)$ with $0 \leq x,y \leq 1$ and $0 \leq x+y \leq 1$ then are $X$ and $Y$ independent?