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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Independence of groups within a single predictor variable

Say I have two variables: 'Sale Price' of a house, and 'Overall Quality' which is an ordinal variable with values between 1 and 10 (1=lowest quality, 10=highest quality). I want to perform an F test ...
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40 views

How do you test for independence of a process?

I'm working on a research project for one of my professors. He wants to know whether a variable that takes on different values over a large period of time (say 1,000,000 different values) is i.i.d. ...
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Does uneven sampling during 3 time periods violate independence for ANOVA?

A continuous variable is measured at 7 AM, 2 PM, and 7PM, over the course of 2.5 months (summer). An animal is being periodically observed, while the site is monitored. The observation periods are for ...
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Accounting for connectedness during the analysis of nodes in a directed acyclcic graph

I am studying a river network and environmental features measured at different stations along the network. Often, one station is higher up the river than another (or multiple others). I can ...
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51 views

Analyzing the statistical independence of a Bernouli sequence

For my research, I need to check whether a sequence of Bernoulli trails (1 as success and 0 as failure) is a Bernoulli process and are statistically independent. As per the following discussion, and ...
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Alternative to Multinomial Logistic Regression when observations are not independent?

I am dealing with an experimental design with a categorical IV and DV: In a linguistic experiment, I asked speakers to form as many sentences as they could using a limited number of words. They were ...
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2answers
52 views

Given $Z\perp X\mid Y$, is it true in general $Var(Z|h(X,Y))=Var(Z|h(c,Y))?$

Given random variables $X, Y, Z$: If $Z\perp X\mid Y$, then I know that $Var(Z|X,Y)=Var(Z|Y)$ But is it still true in general that $$Var(Z|h(X,Y))=Var(Z|h(c,Y))?$$ here $h$ is a real valued function ...
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Number of Head-Tail in N independent coin flips

Consider a sequence of independent tosses of a biased coin at times $t = 0, 1, 2, . . .$. On each toss, the probability of a ’head’ is $p$, and the probability of a ’tail’ is $1 − p$. A reward of one ...
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Is the regressor (sometimes called “independent” variable) actually independent of the response from a probabilistic perspective?

In supervised learning, we refer to the regressors as independent variables and response variables as dependent, but from a probabilistic standpoint, I am having trouble understanding this. To ...
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24 views

Two-dimensional Multinomial distribution and an estimator under the assumption of independence

We have two-dimensional multinomial distribution $Mult(n, p)$, where $p = (p(x, y))_{x \in \mathcal{X}, y \in \mathcal{Y}}$ is a matrix containing probabilities of outcomes of $(X, Y)$: $p(x, y) = P(X=...
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Mathematical explanation for this relationship

I have analyzed some data, and noticed that it fits this relationship quite well: $$P(A|B\cap C) = P(A|B)\frac{P(A|C)}{P(A)}$$ I am wondering what the interpretation of this would be mathematically, e....
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Non-independance of trial likelihoods in a staircase procedure

In psychometrics, we often want to know, for instance, a given participants' perceptual threshold: the intensity of a stimulus that they can detect 50% of the time. It's common to use staircase ...
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Are two random vectors independent if their corresponding components are all independent?

Let $\mathbf{X} = (X_1,\ldots,X_n)$ and $\mathbf{Y} = (Y_1,\ldots,Y_n)$ be random vectors, and let $f_{\mathbf{X}}(x_1,\ldots,x_n)$ and $f_{\mathbf{Y}}(y_1,\ldots,y_n)$ be their respective pdfs or ...
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1answer
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Conditional independence, partial correlation

In my work, I am modelling graphs by measuring the zero- and first-order conditional independence between the variables. That is, if there are three variables, say $A$,$B$ and $C$, an edge between the ...
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How should I test if a categorical variable and a random real vector are independent?

I saw many good answers from how-do-i-test-that-two-continuous-variables-are-independent. Especially, distance correlation seems to be the best to use in a hypothesis test for independence of two ...
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Given samples, how to test an assumption on the distribution that the samples are drawn from?

I have samples from unknown distribution $P(a_1|a_2,b_1,b_2,\dots,b_k)$, which might be independent of $b_1,b_2,\dots,$ or $b_k$ I wonder if there is any hypothesis test that can do the following ...
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What is the significance of “orthogonal” vectors in statistics?

So I am reading What does orthogonal mean in the context of statistics?, and there are contradictory answers. The most upvoted answer says that "Therefore, orthogonality does not imply ...
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1answer
52 views

Compute 16 var(x)+32 var(y) for given bivariate CDF

\begin{equation} {F(x,y)} = \begin{cases} 0 & \text{if $x<0$ or $y<0 $} \\ \frac{1-e^{-x}}{4} & \text{if $x>0, 0 \leq y <1$} \\ 1-e^{-x}& \text{if $x \geq 0, y \...
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60 views

Joint distribution from multivariate Normal distribution

Consider two random variables $\{Y_1, Y_2\}$, which follow a joint multivariate normal distribution: $Y = [Y_1, Y_2]^T,$ \begin{equation} Y_1 \sim \mathcal{N}(\mu_1,\sigma_1),\; Y_2\sim\mathcal{N}(\...
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What does “A realization of independent copies”? [duplicate]

I read in statistic, "a realization of independent copies" from "Elements of copula modelling with R", and do not understand the meaning. I search and found that it means, the two ...
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Prove a result on expectation with 2 random variables

Part 1 I wish to show that: $$E(X)=E[(X|Y=y1)*Pr(Y=y1)+(X|Y=y2)*Pr(Y=y2)]$$ where the random variable Y can take 2 possible values. As many comments and answers have suggested, this may be technically ...
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How to determine the Wald-Wolfowitz Runs Test threshold value for non-normal distributions?

If the underlying distribution is strongly skewed, i.e. Gumbel or Weibel, should the Wald-Wolfowitz Runs Test use that distribution, rather than the Gaussian / Normal distribution, to set the null ...
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Can I use a chi-square test to determine whether my samples are independent or paired?

I am conducting statistical tests for my current research project. Within the scope of this research project, I have two samples (divided by country clusters). The samples contain approximately 1500 ...
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22 views

Definition indepedence and identically distributed (iid)

Bruce Hansen's book "Econometrics" defines a random sample as follows: "The observations $(y_i,x_i, z_i)$ are a random sample if they are mutually independent and identically ...
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Test for independence between variables [closed]

I have the weekly prices of two stocks over a period of 20 years. How could I test the independency between those two stocks?
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Confused about independent probabilities. If a fair coin is flipped 5 times, P(HHHHH) = 0.03125, but P(H | HHHH) = 0.5

I'm confused about how to reconcile the probability of independent events not having anything to do with prior history, but sequences of events do (seemingly) take into account prior history. This ...
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1answer
25 views

difference in CDFs and pdfs of joint distribution of two random variables

We know that the joint probability function of two independent random variables is just the product of their respective pdfs. On the same lines, .can we say that if we multiply the cumulative density ...
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2answers
46 views

Under which conditions does $X_1 \perp X_2, X_3$ imply $X_1 \perp X_2 | X_3$?

E.g. let us imagine we have $X_3 := X_1 \text{XOR} X_2$ with both $X_1, X_2$ being sampled from $\{0, 1\}$ with $p=0.5$. Then $X_1 \perp X_2, X_3$ but $X_1 \not \perp X_2 | X_3$. Are there conditions ...
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Should I use normality or independence test?

I have a datasets with 3 variables and 500 observations. The data is multivariate normal, and the 3 variables are independent. I am doing some 'experiments' with the dataset, where I apply some ...
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39 views

Are these independent random variables?

Let $X, Y, Z$ be pair-wise independent random variables. If $A=XY$ and $B=XZ$, are $A$ and $B$ independent? My thoughts are that they are not independent. For example, if $X$ takes values $-1, 1$ ...
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Mean field variational inference in high dimensions

Is it expected that independence assumption done in mean field variational inference is going to yield on average better approximations for low-dimensional models compared to high-dimensional ones?
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2answers
56 views

uniform distribution, showing that two variables are uncorrelated and not independent

Let (X,Y) have uniform distribution on the four points(0,1),(0,−1),(1,0),(−1,0). How can I show that X and Y are uncorrelated but not independent? Could someone just point me in the right direction ...
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2answers
69 views

PCA vs linear Autoencoder: features independence

Principal component analysis is a technique that extract the best orthogonal subspace in which we can project our points with less information loss, maximizing the variance. A linear auto encoder is ...
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1answer
54 views

Assume $X,Y$ are two independent random variables. Let $Z=f(X,Y)$. If $Z$ is independent of $X$, $f(X,Y)$ is constant in $X$. Is this true?

Let $X\in \mathbb{R}^n$ and $Y\in \mathbb{R}^m$ be two independent random vectors. Then, say that we have a third real valued random variable $Z=f(X,Y)$, with $f$ being measurable. Say that we know ...
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1answer
47 views

How to reason about independence of combinations of events?

Suppose sets $A , B,$ and $D$ are independent. Is it guaranteed that $A \cap B^c \cap D$ is independent from $B^c \cup D^c$? Isn't $B^c$ ($B$ complement, or $B$ not happening) giving me ...
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Notation of independence of random variables

Consider $X = (X_1, X_2, X_3)$ (random variables) with $X_1\sim \mu_1$, $X_2\sim \mu_2$, $X_1\sim \mu_3$ and some function $f$ $$ f = \left\{\begin{matrix} \mathcal{X}\mapsto\mathcal{Y}\\ X \mapsto \...
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Independent or not?

I'm going to analyze the effect of four drugs on a continuous outcome. Each drug has a specific combination of two components with two levels each, i.e. there's one drug for each combination. My ...
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29 views

Independence for subsets of [0,1]

I'm struggling with this problem: Let $\mathbb{P}$ be the uniform measure on $[0,1].$ Define $A=(a,b)$ and $B=(c,d)$ with $a<c.$ State the necessary and sufficient conditions for $A$ and $B$ to be ...
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22 views

Is the independence of this sequence of random variables not implicitly given when we define their probability distributions?

In this post, the user asks whether the following random variable converges to $0$ almost surely: $X_n = \begin{cases} 0, & \text{with probability 1 - $\frac{1}{2n}$,} \\ n, & \text{with ...
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1answer
28 views

correlation of two features, which test? [closed]

I have the following table as pandas dataframe with features feat1 and feat2: ...
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144 views

Is an independent process always a white noise process?

In econometrics, an independent process means that all values are independent of each other, but does this also mean that all independent processes are white noise processes? and is the reverse true?
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1answer
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If a discrete random variable is independent to other two discrete random variable, is it independent to the sum?

Suppose X,Y,Z are discrete random variables and X is independent to both Y and Z. Is X independent to Y + Z? I know this is not necessarily true but I am struggling to find a counterexample. It ...
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Let A and B be two random variables, both independent from another random variable C. Is A*B also independent from C?

Let A and B be two random variables both independent from another random variable C. If A is independent from B, is A*B also independent from C? And if A and B are no independent from each other?
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Boxplot without top whisker

I'm having some questions and hope some one could help me with them. When I plot the data into a boxplot, there is no top whisker but lots of outliers which is above the upper bound. As I'm looking ...
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13 views

How are terms in SS (corrected sum of squares) independent?

What does the highlighted test mean? How are the terms not independent? Can someone elaborate in detail?
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51 views

Are statistics calculated from Bootstrapped samples independent?

I was wondering if someone could point me to some reference stating the (in)dependence properties of statistics calculated from bootstrapped samples. In other words, are bootstrapped statistics (mean,...
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37 views

Are the factors in a time series factor model independent over time?

Consider the time series extension to the standard factor model: $$X_t = \Lambda F_t + e_t, \qquad t = 1, 2, \ldots, T$$ where $X_t$ represents \the vector of observations at time $t$, $F_t$ ...
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33 views

Why do we use the term “uncorrelated” to describe linear dependence/independence?

Terminologically, "uncorrelated" to me means that 2 things have no relationship, not necessarily constrained to linear relationships. However, in statistics, we seem to confine "uncorrelated" to mean ...
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21 views

Doubt about independent and dependent events

For example: An airport screens bags for forbidden items, and an alarm is supposed to be triggered when a forbidden item is detected. Suppose that $5\%$ of bags contain forbidden items. If a bag ...
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40 views

Independence of MLEs of 2 parameter exponential, and showing functions of them are chi-square

Consider a random sample of size $n$ from a two-parameter exponential distribution, $X_i \sim $EXP($\theta,\eta$), and let $\eta^*$ and $\theta^*$ be the MLEs. a) Show that $\eta^*$ and $\theta^*$ are ...

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