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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Casino odds - Independent events and theoretical probability [duplicate]

Let's say you are in a Casino and you are playing Craps. Your desired outcome is 1/36, (every time it is rolled). Every event is independent. If you get a hit (a 1/36 chance) you get paid 36 times ...
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How to determine if two events are independent or not?

I have this question on an ungraded practice test. The possible outcomes of an experiment are 1,2,3,4,5,6,7,8, and assume that all of the outcomes are equally likely. Let A be the event that the ...
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How large a dataset do you need to use distance correlation?

How large a dataset do you need to use distance correlation? Distance correlation is well explained in Stackexchange post: Understanding distance correlation computations What is not clear to me is ...
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Independence/ Asymptotic independence of asymptotic normal random variables

Let $\{X_{n}\}_{n\in I}$ be a sequence of random variables, where $X_{n}$ takes value $\{-c_n,c_n\}$, each with probability $1/2$, $|c_{n}|\leq \alpha \in \mathbb{R}$ and $I$ denotes the index set ...
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When do we find convergence in distribution to independent variables?

Let $\{X_{i}\}_{i=1}^{n}$ and $\{Y_{i}\}_{i=1}^{n}$ are two sequences of random variables such that $\bar{X} = \sum_{i=1}^{n} X_{i}$ and $\bar{Y} = \sum_{i=1}^{n} Y_{i}$ asymptotically converges in ...
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Informative Censoring vs. Random Censoring vs. Conditionally Independent Censoring

Let us consider the case of survival analysis with one event. Let $X$ represent a set of covariates about each unit. Let $T_E$ be the (latent) event time of the unit, let $T_C$ be the (latent) ...
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Confused with independence and Bayes theorem

I am doing one calculation in two different ways and the result does not match. Let two sets of random variables $\bar{y} = (y_1, y_2, \ldots, y_N)$ and $\bar{x} = (x_1, x_2, \ldots, x_N)$. Each $y_i$...
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IV estimation for when instrument is independent of the outcome

So I came across this post and have some use for it in my own work. Recap of Original Question: In the canonical IV setting, suppose we have a data-generating model $Z\rightarrow X\rightarrow Y$ (...
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Proving independence of discrete variables and the product of them

Given that P(A) and P (B) are independent and \begin{equation} $P(A=1)=\frac{1}{2}$ $P(A=-1)=\frac{1}{2}$ $P(B=1)=\frac{1}{2}$ $P(B=-1)=\frac{1}{2}$ There is a random variable $C = A \cdot B$, C is ...
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Binomial vs Bernoulli Likelihoods: Difference assumptions of independence across observations

In Bayesian statistics, logistic regression can be facilitated by priors, a link function, and a likelihood choice of either the Bernoulli or Binomial distributions. My question is whether this design ...
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How to include an independent variable as an input to a statistical model

I want to create a statistical model of the following model inputs $v_{in}$ and outputs $v_{out}$ that consist of continuous random variables. The final aim is to calculate the conditional ...
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What is the distribution of the sum of differences of i.i.d. variables?

Let's suppose that $X, X_1, X_2, \ldots, X_n$ are i.i.d. continuous random variables. Let's define $D_i$ as the difference between $X_i$ and $X$: $$ D_i = X_i - X $$ My end goal is to derive the ...
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Dependence in probabilistic machine learning

From this slide: slides page 39, it says: In probabilistic machine learning prediction time: compute the likelihood given parameters, each parameter configuration of which is weighted by the posterior:...
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$X+Y=s$ but $X$ and $Y$ are independent?

From another Cross Validated question Given $X\sim N(\mu_X, \sigma_X^2)$ and $Y\sim N(\mu_Y, \sigma_Y^2)$ are independent, and you know $X+Y=s$. What is the expected value of $X$? The question makes ...
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Predictive modelling strategy for spatial interpolation: complex data structure, small sample size, but n > p

I apologise in advance for the long post, my questions are deeply interconnected so it felt wrong posting them as separate threads. Please note also that I edited this post to account for Florian ...
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Show that intersection property - $I(X,Y\cup Z, W)$ and $I(X,W\cup Z,Y)$ $\implies$ $I(X,Z,Y\cup W)$ - requires strict positive distribution

Question It is stated in Probabilistic Reasoning In Intelligent Systems by Judea Pearl that, intersection property of information relevance axioms - $I(X,Y\cup Z, W)$ and $I(X,W\cup Z,Y)$ $\implies$ $...
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Independence-violation issue in pairwise measurements: Which test?

Here is another problem in which independence might be an issue (however, the question might be very basic). Suppose, I have a rowing team and I have two boats (B1 and B2, for brevity). I have strong ...
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Example of two dependent RVs where all subsets of first and second RV has no correlation?

From Simple examples of uncorrelated but not independent 𝑋 and 𝑌, we have: 𝑋 ∼ 𝑈(−1,1) <...
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Random effect for handling multiple (but varying) outcomes per individual

I have a dataset where I have at least one clinical measurement for 1000 individuals, but for a subset I have more than measurement. These are patients with a disease that can temporally become more ...
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Independence in Graphical model of $p(h_{1:T}|v_{1:T})$ of an HMM

I am studying Hidden Markov Models and I'm trying to understand the following exercise: Consider Hidden Markov Model with hidden states $h_{1:T} = \{h_1,...,h_T\}$ and observed states $v_{1:T}=\{v_1,.....
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$cov(X,f(X))\neq 0$ and $E(X f(X))\neq 0$

Take a random variable $X$. Is it true that (1) $cov(X,f(X))\neq 0$ for any function $f$? (2) $E(X f(X))\neq 0$ for any function $f$? I believe the answer to both questions is no. However, can you ...
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Enforcing independence of time to event and censoring time in survival analysis

I am studying survival analysis and reading Tutz & Schmid "Modeling Discrete Time-to-Event Data" (2016). The authors often consider the censoring time $C$ to be independent of the time ...
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Clarify, with example, completeness conjecture by Pearl and Paz

I was going through Probabilistic Reasoning In Intelligent Systems by Judea Pearl. A completeness conjecture (for which no complete proof is there as yet, but has been found to be true generally, as ...
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Can all types of probabilistic independencies be depicted via graphs?

I was going through Probabilistic Reasoning In Intelligent Systems by Judea Pearl .In chapter 3 the author tries to motivate the need for qualitative representation of independence relations, that do ...
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Expression for $p(\mathbf{s}|\mathbf{d})$ and the respective Markov Network

I would like to check if I correctly derived my the expression for $p(\mathbf{s}|\mathbf{d})$. Here's the question: Consider a model of diseases and symptoms. $s_i\in\{0,1\}$ is a binary random ...
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Quick question about density with respect to product measure

Suppose we got 2 non indepenendent random variables $X_1:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_1,\mathcal{B}_1)$ and $X_2:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_2,\mathcal{B}_2)$ with ...
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Supra-additive relations among timeseries (separate ratings of a single stimulus)

I conducted an online behavioural study (N=20) where people listened to a song with lyrics, and rated (in three counter-balanced conditions) the tension they perceived in the music alone (M), the ...
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How to get to the formula of this joint probability

In Bishops book, "Statistical Pattern Recognition", in Chapter 8 about Graphical models he introduces the following 3-node graph: The goal is to investigate the independence properties of ...
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Quantifying Dependency in Chi Squared Test

I am running a Chi Squared test on nominal values and this is the contingency table. ...
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Examples of orthogonal but correlated random variables and viceversa

I am trying to understand the difference between orthogonality and zero correlation. There are many insightful questions/answers on the topic in this forum, but I'm looking for some practical examples....
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Which of the following is an independent event? , or are there different types of independent events?

This is a high school math question: There are 6 balls in a bag, and one of them is red, and it is marked #1, 2 yellow ones are marked #1 & #2, 3 blue ones are marked #1, #2, #3 respectively。...
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Quick way to determine the different independence assumptions

This question is different than my previous question in that I'm asking sort of a "meta" question. Here's two graphical models (a Belief Network and a Markov Network): I would like to ...
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Determining unconditional independence in Markov Networks

I would like to know whether $E \perp\kern-5pt\perp A $ in the following Markov Network and would like to know if my reasoning is correct: So, since this is a Pairwise Markov Network, it factorizes ...
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Checking for conditional independence in graphical models

I would like to know whether $B \perp\kern-5pt\perp C | D,A $ and $D \perp\kern-5pt\perp A | B,C $ in the following two graphical models and would like to know if my reasoning is correct: For the ...
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Is $C \perp\kern-5pt\perp D | A $ for the two graphical models? [duplicate]

I would like to know whether $C \perp\kern-5pt\perp D | A $ in the following two graphical models and would like to know if my reasoning is correct: For the left model (Belief Network), here's my ...
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Is $B \perp\kern-5pt\perp C | A $ for the two graphical models?

I would like to know whether $B \perp\kern-5pt\perp C | A $ in the following two graphical models and would like to know if my reasoning is correct: For the left graphical model, which is a Belief ...
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Implications of mean independence

Let $U$ be a random variable with mean $0$. Take other two random variables $X,Y$. Assume $$ (1)\quad E(U|X,Y)=0. $$ I believe (1) implies $$ E(U\cdot X)=E(U \cdot Y)=0. $$ Does (1) imply $$ E(U \cdot ...
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Why the columns of design matrix for linear models with additive Gaussian noise linear independent?

This question comes from page 142 of the book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. Since it takes pages to arrive at the result, I excerpt the major settings ...
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Steiger's Z test, independent samples required?

I have some samples and for each one, I have 3 measures : x, y and z. I want to know if the correlation between x and y and the correlation between y and z are significantly different. I will use ...
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Understanding an expression of probability by the nth fixation

There is a quality control inspector who searches for faults in newly produced items in a series of independent "fixations". If p is the probability that a flaw is detected during any one ...
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What does it mean for a pair of data points to be i.i.d?

I am studying conformal prediction where they mention that the data points in the calibration data should be i.i.d. Here's what it reads - we reserve a moderate number (e.g., 500) of fresh i.i.d. ...
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What is the probability distribution of summation of two independent random variables conditioned upon one of the summands?

What would be $P(\Theta + W \mid \Theta)$ where $\Theta \perp W$. Intuitively I think It should be $P(W)$ shifted by $E[\Theta]$. Though I am not at all sure If we condition it upon a specific $\theta$...
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Are these two events independent (Comey and Mccabe tax returns)? How would we tell?

The NYT reports: The former F.B.I. director and his deputy, both of whom former President Donald J. Trump wanted prosecuted, were selected for a rare audit program that the tax agency says is random. ...
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Probability that two specific people are selected for IRS audit within 4 year period

A recent NYTimes article states that The minuscule chances of the two highest-ranking F.B.I. officials — who made some of the most politically consequential law enforcement decisions in a generation —...
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Does dependence between A & B imply dependence of !A & !B? [duplicate]

If events A&B are dependent, can we say with certainty that !A & !B (the converses of A and B) are also dependent? My intuition tells me that this statement is true, but how would we go about ...
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hypothesis testing with data that are not equally independent

I think the easiest way to explain my question is with an example scenario: Let's say we have 10 groups of 5 people and each group is in an identical circular room in which we are allowing them to ...
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Prove sample minimum and unbiased sample variance are independent

I have a simple random sampling (n observations) of the population $X$, whose pdf is $$f_\theta (x) = e^{-(x-\theta)} I_{(\theta,\infty)}(x) $$ and $\theta \in \mathbb{R}$. $S^2_n$ is the unbiased ...
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Accumulation of evidence in basic statistics

Imagine I have a coin, and I want to study tossing outcomes of that coin. I do this over two experiments. In the first experiment, I toss the coin $n_1$ times and write down all the outcomes, in the ...
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Can I consider the data of various different execution of my agent independent?

I developed an "intelligent agent" and I want to justify the development choices using statistical analysis. My system can be ultimately summarized as a function returning a value composed ...
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How to model multi-level data to interpret features of individuals?

I am currently very thorn about how to best proceed on the problem I am working on. The data consists of observations (y) of the effect of As on Bs. In particular, y is a percentage difference so the ...

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