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.

125 questions
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Can one infer independence by simple reasoning/intuition?

From my recent experience in probability, it feels as though independence is something we "discover" from the system via the equation: $$P(A)*P(B)=P(A\cap B)$$ Could one ever conclude independence ...
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A criterion for independence based on Characteristic function

Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be ...
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A simple way to obtain $\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s}$.

Let $p_1<p_2 <\cdots <p_k < \cdots$ the increasing list in set $\mathbb{P}$ of all prime numbers . By sum of infinite geometric series we have $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$, ...
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Questions on Kolmogorov Zero-One Law Proof in Williams

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Williams' Probability book: Here are my questions: Why exactly are $\mathfrak{K}_{\infty}$ and $\mathfrak{T}$ ...
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Let $U=\operatorname{min}\{X,Y\}$ and $V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$.

Let $X$ and $Y$ be exponentially distributed random variables with parameter $1$ and let $U=\operatorname{min}\{X,Y\}$ and $V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$. We ...
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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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Existence of independent and identically distributed random variables.

I often see the sentence "let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with a certain distribution". But given a random variable $X$ on a probability space $\Omega$, how do I know ...
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Prove that $\mathbb P(X>Y) =\frac{b}{a + b}$ if $X, Y$ are exponentially distributed with parameters $a$ and $b$.

Let $X, Y$ be an exponentially distributed random variables with parameters $a, b$. Then $X$ has pdf: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
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Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent [duplicate]

Let $X$ and $Y$ be independent random variables. Prove that $f(X)$ and $g(Y)$ are independent for any choice of measurable functions $f$ and $g$. This sounds very obvious, but I have no idea how to ...
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$X$,$Y$,$Z$ mutually independent implies $X+Y$ independent of $Z$

Supposing $X$, $Y$ and $Z$ and mutually independent real random variables, how can we prove that $X+Y$ and $Z$ are independent from the definition? If not from the definition, using $\sigma$-algebras? ...
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 $S < 1 \to \prod_{n=1}^{\infty} (1-p_n) \geq 1 - S$. ...