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.

4
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3answers
292 views

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 ...
12
votes
2answers
5k views

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 ...
27
votes
2answers
1k views

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 $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$, ...
5
votes
1answer
2k views

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 $\mathfrak{T}$ ...
5
votes
3answers
1k views

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 ...
2
votes
1answer
443 views

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 ...
23
votes
3answers
2k views

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 ...
5
votes
2answers
606 views

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: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
1
vote
1answer
398 views

Prove symmetry of probabilities given random variables are iid and have continuous cdf

Let $Y_1, Y_2, ...$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_{Y}(y) := F_{Y_1}(y) = F_{Y_2}(y) ...
0
votes
1answer
144 views

Choosing the correct subsequence of events s.t. sum of probabilities of events diverge

Here is the problem. I tried choosing $B_n = A_{mn}$ since it is an independent sequence for $m \geq 2$, but I am not quite sure how to guarantee that $\sum_{n=1}^{\infty} P(A_{mn}) = \infty$. Is it ...
0
votes
1answer
199 views

Mutual Independence Definition Clarification

Let $Y_1, Y_2, ..., Y_n$ be iid random variables and $B_1, B_2, ..., B_n$ be Borel sets. It follows that $P(\bigcap_{i=1}^{n} (Y_i \in B_i)) = \Pi_{i=1}^{n} P(Y_i \in B_i)$...I think? If so, does ...
13
votes
2answers
5k views

Independence of disjoint events

I'm taking a class in Probability Theory, and I was asked this question in class today: Given disjoint events $A$ and $B$ for which $$ P(A)>0\\ P(B)>0 $$ Can $A$ and $B$ be independent? My ...
-1
votes
2answers
2k views

If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If no:...
5
votes
3answers
2k views

Dependence and second Borel-Cantelli lemma.

I'll put the problem and then I'll explain my problem. Problem: Let ${A_n}$ be events such as $\operatorname{Cov}(I_{A_i},I_{A_j})=E[I_{A_i}I_{A_j}]-E[I_{A_i}]E[I_{A_j}]\leq 0,\ \forall i\neq j\tag{...
4
votes
2answers
733 views

Durrett Example 1.9 - Pairwise independence does not imply mutual independence?

The example in question is from Rick Durrett's "Elementrary Probability for Applications", and the setup is something like this: Let $A$ be the event "Alice and Betty have the same birthday", $B$ ...
2
votes
1answer
1k views

Maximal inequality for a sequence of partial sums of independent random variables [closed]

Let $X_n$, $n=1,2,3,...$ be a sequence of independent (not necessarily identically distributed) random variables, let $S_n=\sum_{i=1}^nX_i$. Prove the following maximal inequality: For all $t>0$,$$\...
3
votes
1answer
6k views

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 ...
1
vote
1answer
891 views

$X$,$Y$,$Z$ 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? ...
-3
votes
1answer
531 views

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 $S < 1 \to \prod_{n=1}^{\infty} (1-p_n) \geq 1 - S$. ...
11
votes
4answers
2k views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
6
votes
2answers
2k views

Product of independent random variables

The following is a classic example that pairwise independent does not necessarily imply mutually independent: Let $X_1$ and $X_2$ be independent r.v.'s with distributions $$P(X_i=1)=P(X_i=-1)=\frac{...
4
votes
1answer
258 views

Given $U$ with known PDF, find $W$ independent of $U$ such that $U+W$ is distributed like $2U$

Let $U$ denote a random variable with PDF $$f_U(u)=ce^{-u\sqrt{u}}\,\mathbf 1_{u>0}$$ Does there exist a random variable $W$ independent of $U$ such that $U+W$ is distributed like $2U$? This ...
4
votes
2answers
911 views

Independent $\sigma$-algebras using $\pi$-$\lambda$-theorem

Let $\mathcal{E}_1, ...,\mathcal{E}_n$ be collections of measurable sets on $(\Omega,\mathcal{F},P)$, each closed under intersection. Suppose \begin{align*} P(A_1\cap...\cap\ A_n)=P(A_1)\cdot ... \...
3
votes
1answer
607 views

$\sigma$-algebra of independent $\sigma$-algebras is independent

Let $\{\mathcal{A}_i:i \in \mathcal{I}\}$ be a collection of independent $\sigma$-algebras. Let $I_1, \ldots , I_n$ be pairwise disjoint subsets of $\mathcal{I}$ and let $\mathcal{B}_k = \sigma\{\...
2
votes
1answer
248 views

“Fair” game in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. What's fair about a fair game? Let $X_i$, i = 1, 2, ... be indp RVs s.t. $X_i = i^2 - 1$ with prob $1/i^2$ and $-1$ with ...
9
votes
3answers
4k views

Probability sum of 5 before sum of 7

Pair of fair die are rolled (independently I hope) infinitely. Find probability sum of 5 appears before sum of 7. 2 approaches: $$P(\text{sum of 5 appears before sum of 7})$$ $$= P(\text{roll 1 is ...
3
votes
2answers
217 views

Prove independence of events given random variables are iid and have continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_Y(y) := F_{Y_1}(y) = F_{Y_2}(y)...
1
vote
1answer
396 views

Questions on Kolmogorov Zero-One Law Proof in Rosenthal

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Rosenthal's Probability book: Here are my questions: Question 1: In the first red box, does the fact that Q and P ...
1
vote
1answer
138 views

How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$?

Suppose we have a time series $X_t$ s.t. $X_t \sim^{iid} (0,1)$. How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$? Or, I guess, if $X,Y\sim^...
78
votes
5answers
42k views

Could someone explain conditional independence?

My understanding right now is that an example of conditional independence would be: If two people live in the same city, the probability that person A gets home in time for dinner, and the ...
10
votes
3answers
875 views

Examples of pairewise independent but not independent continuous random variables

By considering the set $\{1,2,3,4\}$, one can easily come up with an example (attributed to S. Bernstein) of pairwise independent but not independent random variables. Counld anybody give an example ...
9
votes
4answers
1k views

Showing that $n$ exponential functions are linearly independent.

I have $n$ lambdas, which are all different real and positive numbers, where: $\lambda_1 < \lambda_2 < \cdots < \lambda_n$. I then have to show that these functions are linearly independent:...
2
votes
0answers
288 views

Converse of Kolmogorov's Zero-One Law

The Kolmogorov zero-one law says that for a sequence of independent events, any event belonging to the tail $\sigma$-field has probability either $0$ or $1$. However the converse is not true, because ...
3
votes
1answer
123 views

A variant of Kac's theorem for conditional expectations?

This is part of the proof of the Strong Markov property of Brownian motion given in Schilling's Brownian motion. Here $B_t$ is a $d$-dimensional Brownian motion with admissible filtration $\mathscr{...
2
votes
1answer
725 views

Martingale formulation of Bellman's Optimality Principle

Related question: Deducing an optimal gambling strategy (using martingales). What I tried: For no 2, if $\ln Z_n - n \alpha$ is a supermartingale, then for $m < n$, $$E[\ln Z_n - n \alpha | \...
2
votes
1answer
404 views

Assume $A,B,C,D$ are pairwise independent events. Decide if $A\cap B$ and $B\cap D$ are independent

Assume A,B,C,D, are pairwise independent events. Decide if $(A \cap B)$ and $(B \cap D)$ are independent events? Then repeat this assuming the four events are mutually independent. Well, what I'm ...
1
vote
1answer
103 views

Use inclusion-exclusion to find explicit formula for $P(A_k)$. Where is independence used?

Probability with Martingales This was answered here using bounds and here using PGF. I would like to try inclusion-exclusion. Let $H_1, H_2, ...$ be independent events with $P(H_k) = p$ where $H_k$ ...
7
votes
1answer
3k views

Independence and conditional expectation

So, it's pretty clear that for independent $X,Y\in L_1(P)$ (with $E(X|Y)=E(X|\sigma(Y))$), we have $E(X|Y)=E(X)$. It is also quite easy to construct an example (for instance, $X=Y=1$) which shows that ...
5
votes
1answer
694 views

Conditional independence given the complement of an event

Suppose $A_1,\ldots A_n$ are conditionally independent given $B$. Are they conditionally independent given $B^c$ as well?
3
votes
3answers
133 views

Does $P(A) = 1$ imply that $P(A) = P(A \mid B) = P(A \mid \neg B) = 1$?

Suppose for proposition $A$ we have that $$P(A) = 1$$ Then does it follow that for all $B$ $$P(A) = P(A \mid B) = P(A \mid \neg B) = 1?$$
2
votes
0answers
25 views

Simplify $\int_{\Omega} \int_{\Omega} 1_{A}(\omega) g(\tilde{\omega}, \Pi(\omega)) d\mathbb{P}(\tilde{\omega}) d\mathbb{P}(\omega) $

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a (complete) probability space and $D$ be a compact topological space, equipped with its canonical Borel $\sigma$-algebra $\mathcal{B}(D)$. Furthermore, ...
2
votes
0answers
296 views

Kolmogorov 0-1 Law Converse?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1$. There ...
3
votes
1answer
164 views

Prove independence of a pairwise independent subsequence of independent events

Consider infinite independent coin tossing where $H_n = \{$nth coin is heads$\}$ for $n = 1, 2, ...$. Let $$A_n = \bigcap_{i=1}^{\left \lfloor \log_2 n \right \rfloor} H_{n+i}$$ How do you show that ...
3
votes
1answer
393 views

Find three Poisson-distributed random variables, pairwise independent but not mutually independent

I am asked to give an example of three Poisson-distributed random variables which are pairwise independent, but are not mutually independent. I thought of the example about the Intersection where ...
2
votes
1answer
45 views

Why $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent if $X_1$ and $X_2$ are i.i.d. exponential?

How to show that $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent, if $X_1$ and $X_2$ are i.i.d. exponential with mean $1$? Is there a simple way to see this?
2
votes
1answer
117 views

If $X$ and $Y$ are independent $N(0,\sigma^2)$, then $X^2+Y^2$ and $X/Y$ are independent?

If $X$ and $Y$ are independent, then $X^2+Y^2$ and $X/Y$ are independent? I was solving the problem for the case that $X$ and $Y$ are independent $N(0,\sigma^2)$. So i found that $X^2+Y^2$ is ...
2
votes
1answer
133 views

Pairwise Independence of Events in Rosenthal [duplicate]

$\exists$ this exercise in Rosenthal's A First Look at Rigorous Probability Theory: For letters d and e, how do you show that the ff events are pairwise independent? My attempt: It suffices to show ...
2
votes
2answers
905 views

Can linearly independent vectors have linearly dependent images?

If a two vectors $\bf u$ and $\bf v$ in linearly independent, can their images be linearly dependent given they were transformed by a linear transformation $T$? I think not. My reasoning If the ...
1
vote
2answers
47 views

$X_1,…,X_n$ be independent RVs and $X_i \perp \mathcal F $ for $1\ \leq \forall i \leq n$. Show that $\sigma (X_1,…,X_n) \perp \mathcal F$

Let $X_1,...,X_n$ be mutually independent RVs. Suppose $X_i \perp \mathcal F $ for $1\ \leq \forall i \leq n$. How can I show that: $$\sigma (X_1,...,X_n) \perp \mathcal F$$ ? What I have tried: ...
0
votes
1answer
117 views

Prove $Y_0$ is $\mathscr{L}$-measurable.

Let $Y_0, Y_1, \ldots$ be independent random variables with $P(Y_n = 1) = P(Y_n = -1) = 1/2$ for $n = 0, 1, 2, \ldots$ Define $X_n = Y_0Y_1Y_2\cdots Y_n = \prod_{i=0}^n Y_i$ for $n = 0, 1, 2, \ldots$...