# Questions tagged [independence]

For questions involving the notion of independence of events, of independence of collections of events, or of independence of random variables. Use this tag along with (probability) or (probability-theory). Do not use for linear independence of vectors and such.

1,445 questions
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### Condtional Covarianz Problem

i have on Friday a presentation and i have one small problem. Let $X_1,...,X_n$ be independent Random Variables. Let $f: \Omega^n \rightarrow \mathbb{R}$ and define $Z:=f(X_1,...,X_n)$. Define also: \...
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### Bernoulli vs. binomial: Which one to apply in practice?

After reading this and this thread, I got a little confused about the distinction between the Bernoulli and the binomial distribution in practice. From what I read, I understood that: A trial (e.g. ...
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### Infinite sums of independent random variables

Question: Suppose we have an independent family of random variables $$\{X_{ij} \mid i \in \{1,\dots,m\}, j \in \mathbb{N} \}$$ on the same probability space $(\Omega, \mathcal{A}, \mathbb{P})$. ...
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### Linear algebra questions: True or False [on hold]

5 vectors in $\mathbb{R}^6$ are always dependent? If $A$ is singular $n \times n$ matrix, $A^T A$ is also singular? If $P$ is a permutation matrix, then $P$ must be singular? A remedy for the accurate ...
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### Are the row vectors in a row reduced echelon matrix always independent?

Are the row vectors in a row reduced echelon matrix always independent? I'm thinking that since the first row is the only row with a non-zero coefficient, then it must be independent of all the ...
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### Prove expectation of independent R.V.s. are independent

Let $X_1, X_2, \dots$ be independent random variables, and show that $Y_n = X_n - \mathbb{E}[X_n]$ are independent
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### Need help fixing/clarifying my thinking about iid RVs after learning some 1st Yr Stats

First a warning: this is not the most interesting question but I want to update my understanding of independence now that I'm taking 1st year statistics I often heard in my 1st year probability class ...
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### Example of an experiment in which A, B, C are independent, but not pairwise independent

Can somebody give an example of process in which we have at least three events A, B, C and: P(A ∩ B ∩ C) = P(A) * P(B) * P(C) But A, B, C are not pairwise independent
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### Conditional i.i.d.ness of random variables

Let $X$ be a random variable with support $\mathcal{X}$. Let $r:\mathcal{X}\to\{0,1\}$. Assume the random variables in the sequence $(\epsilon_i)_{i=1}^N$ are i.i.d. conditional on $X$. Does this ...
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### If $X$ is inpendent of $\mathcal{F}$, then why is for $A \in \mathcal{F}$, $\chi_{A}$ and $X$ independent

As the title suggests, let $X$ be a real-valued random variable and $\mathcal{F}$ a given $\sigma-$algebra. Let $A \in \mathcal{F}$, then it immediately follows that $\chi_{A}$ and $X$ are ...
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### How to understand and to prove that any $n$ linearly independent vectors form a basis of $R^n$ space conceptually?

I know how to use methods like system of linear equations to show that any vector in $R^n$ can be expressed in the form of unique linear combinations of a given set of n linearly independent vectors, ...
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### Two vectors are linearly independent?

Let $x, y, z$ be vectors in vector space $V$. Suppose $z \notin L(x,y)$ , where $L(x,y)$ is the linear span of $x, y$. Show that $x, y$ are linearly independent iff x+z, y+z are linearly independent. ...
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Let $(X_{n})_{n \in \mathbb N}$ be a Markov Chain How can I show $\sum\limits_{y \in E}\sum\limits_{n \geq 0}P^{x}(\tau_{x}>n,X_{n}=y)\Pi(y,z)=\sum\limits_{n \geq 0}\sum\limits_{y \in E}P^{x}(\... 1answer 22 views ### Bounds for the number of independent variables on a finite probability space The following question looks somehow academic but I didn't find much information on the Internet. Consider a finite probability space$\Omega=\{\omega_1, \dots, \omega_n\}$with a probability$\...
I am struggling to work through the following problem. Let $\{X_{n}\}_{n\geq 0}$ be i.i.d. random variables, $M$ a Poisson random variable with parameter $\lambda$ that is independent of the random ...