# 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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### $cov(X,f(X))\neq 0$ 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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### 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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