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.

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Probability of “Not A or Not B” when events are independent [closed]

Let's say $P(A) = .5$ and $P(B) = .2$ and they are independent events. What would $P(A^{c}\cup B^{c})$ be? Any help will be appreciated!
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Show the probability of liminfA is 1

Consider a random set $S$ constructed as follows. For any $n>2$, $\mathbb p(n\in S) = 1/\log n$, all independently. Show that $S$ almost surely satisfies the twin prime property: there ...
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Verifying Linear Independence

I saw the following question from M. Artin's book, Algebra. I simply need to show the the functions $\cos x$, $\sin x$ and $e^x$ are independent. I have no idea how to show their independence. Any ...
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Is the subset $\{f_n: n\geq 0\}$ of $\mathbb{R}^\mathbb{R}$ linearly independent.

For each non-negative integer $n$, let $f_n\in \mathbb{R}^\mathbb{R}$ be the function defined by $f_n$: $x\to \sin^n(x)$. Is the subset $\{f_n: n\geq 0\}$ of $\mathbb{R}^\mathbb{R}$ linearly ...
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Show that the set $K(v,{1\over 2}(w+y))\cap K(w,{1\over 2}(v+y)) \cap K(y,{1\over 2}(v+w))$ is nonempty, and determine how many elements it can have.

Let $V$ be a vector space over $\mathbb{R}$. For vectors $v\neq w$ in $V$, let $K(v,w)$ be the set of all vectors in $V$ of the form $(1-a)v+aw$, where $0\leq a\leq 1$. Given vectors $v,w,y\in V$ ...
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Prove $S \doteq \sum_{n=1}^{\infty} p_n < \infty \to \prod_{n=1}^{\infty} (1-p_n) > 0$ assuming $0 \leq p_n < 1$. [duplicate]

Prove $S \doteq \sum_{n=1}^{\infty} p_n < \infty \to \prod_{n=1}^{\infty} (1-p_n) > 0$ assuming $0 \leq p_n < 1$. Hint: Show that $S < 1 \to \prod_{n=1}^{\infty} (1-p_n) \geq 1 - S$. ...
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Borel-Cantelli-related exercise: Show that $\sum_{n=1}^{\infty} p_n < 1 \implies \prod_{n=1}^{\infty} (1-p_n) \geq 1- S$.

This is supposed to be related to the 2nd Borel-Cantelli Lemma (my justification for the independence tag). In Williams' Probability with Martingales, 2BCL is proven and then the following is given as ...
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Completions of $\sigma$-algebras generated by Levy process are independent

This question arose from attending a seminar about stochastic processes using the book "Introduction to the Theory of Random Processes" by N.V. Krylov. I'm not gonna follow the notation of the book ...
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Probability and Independent Events

A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase ...
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Non-independence of three events, given their intersection is independent

Suppose you flip a fair coin three times, find three events $A$, $B$, and $C$, such that no two of them are independent, but $P(A \cap B \cap C) = P(A)P(B)P(C)$ I am given the question above. ...
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Flipping a fair coin three times: How to find independent/dependent events?

Given that I have a fair coin which I toss three times, I have the following sample space: $S=\left\{HHH , HHT , HTH, THH , HTT, TTH, THT , TTT\right\}$ how do I: a) Find three events A , B, and C ...
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Is linear independence preserved through column transformations?

Let's say you have a $m\times n$ matrix $A$, and the $n$ column vectors are linearly independent. And let's say you have a transformation $T$. You perform the transformation on each column of $A$, ...
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How to use induction to show that $\delta(\mathcal G_1), \ldots, \delta(\mathcal G_n)$ are independent?

I have proven that if the systems $\mathcal G$ and $\mathcal H$ are independent then so are the Dynkin systems $\delta(\mathcal G)$ and $\delta(\mathcal H)$. Now I'd like to generalize it to $n$ ...
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If independent r.v. converge in probability to a constant, do they converge almost surely?

I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and ...
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Continuous distribution and independence [closed]

Problem: In a room, there are 4 boys from high income families, 6 girls from high income families and 6 boys from low income families. How many girls from low income families also need to be present ...
What exactly is $\cap$-stable here?
From my lecture notes: Theorem: Let $(\Omega, \mathcal A, P)$ be a probability space, $A \in \mathcal A, \mathcal M := \{ M_1, \ldots, M_n \} \subset \mathcal A$. The following statements are ...