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!
3
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0answers
102 views

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 ...
2
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0answers
365 views

An equivalent of a central limit theorem for a geometric mean

I'm taking a probability theory actuary exam next week and came accross an interesting problem - consider a sequence $(X_n)_{n=1}^{\infty}$ of i.i.d. random variables with density $$f(x) = \frac{3}{x^...
2
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2answers
1k 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 ...
0
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1answer
240 views

Prove random variable $Y_0$ and $\sigma$-algebra $\mathscr{R}$ are independent. [closed]

Let $Y_0, Y_1, ...$ be independent and identically distributed random variables with $P(Y_n = 1) = P(Y_n = -1) = 1/2$ for n = 0, 1, 2 ... Define random variables $X_n = Y_0Y_1Y_2...Y_n = \prod_{i=0}^...
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1answer
119 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 $n = 0, 1, 2, \ldots$...
1
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1answer
53 views

Prove that $T_{1}, T_{2}-T_{1}, \dots$ are independent

Let $X_{n}$ be a sequence of independent random variables following a Bernoulli distribution of parameter $p$. We define $T_{0} = 0$ and $\forall n \geq 1,$ $$T_{n} = \inf \lbrace k > T_{n-1} ; ...
1
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3answers
64 views

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 ...
1
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1answer
62 views

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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1answer
29 views

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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1answer
574 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 $S < 1 \to \prod_{n=1}^{\infty} (1-p_n) \geq 1 - S$. ...
2
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1answer
464 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 ...
1
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1answer
87 views

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 ...
0
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1answer
300 views

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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2answers
775 views

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. ...
2
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1answer
595 views

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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0answers
55 views

$\sup B_{1 + t} - B_1$ is independent of $B_1$ for $B$ a brownian motion

Is $\sup_{t \ge 0} B_{1 + t} - B_1$ is independent of $B_1$ for $B$ being a brownian motion? For all $t$ $B_{1+t} - B_1$ is independent of $B_1$ but this isn't enough. I also can't use that $\sup_{t \...
2
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1answer
452 views

Two uncorrelated random variables both taking only two values are independent

Let X and Y be random variables both taking only two values. Show that if they are uncorrelated then they are independent.
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2answers
742 views

Proof that $2^n-(n+1) $ equations are necessary to establish the independence of n events.

Suppose $A_1,A_2,\cdots,A_n$ are $n$ events, we say that they are all independent if for all $\{i_1,\cdots, i_m\}\subset \{1,2,\cdots,n\}$(where $m\ge 2$), we have $$\mathrm{Pr}[A_{i_1}\cap A_{i_2}\...
2
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1answer
383 views

Show $X$ and $Y$ are independent if we assume that $E[XY] = E[X] E[Y] $

Assume that $$E[XY] = E[X]E[Y]$$ Let $X$ and $Y$ be random variables taking two different values $a,b \in \mathbb{R}$. Show that X and Y are independent. Note: I've spent a long time on this problem....
2
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1answer
511 views

Finding the distribution of the sum of n independent random variables having exponential distributions

My graduate level probability class asks us to calculate the distribution of the sum of n independent exponentially distributed random variables. I am trying to perform many convolutions but it gets ...
6
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0answers
152 views

Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R $ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
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1answer
2k views

Properties of independence and conditional independence

Recently, I see some properties from conditional independence wiki page https://en.wikipedia.org/wiki/Conditional_independence I don't quite understand the properties of "Rules of conditional ...
6
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1answer
167 views

Almost sure convergence through subsequences

$\{X_i\}$'s are independent Poisson random variables with parameters $\lambda_i$ respectively satisfying $\sum_{n=1}^{\infty}\lambda_n=\infty$. Define $S_n=X_1+X_2+\cdots +X_n$ then show that $$\frac{...
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0answers
76 views

Independence of Events in Lovasz Local Lemma

Let $G$ be a (finite) graph with maximum degree $d$ and vertices $v_{1}, \dotsc ,v_{n}$. Let us associate an event $A_i$ with $v_i (i = 1, . . . , n)$ and suppose that $A_i$ is independent of the set $...
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3answers
6k views

Finding the Marginal Distribution of Two Continuous Random Variables

The continuous random variables $X$ and $Y$ have the joint probability density function: $$f(x, y)= \begin{cases} \dfrac{3}{2}y^2, & \text{ where } 0\leq x \leq 2 \text{ and } 0 \leq y \...
2
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1answer
55 views

If $X_n$ are symmetric around $0$ then show $P(|S_n|\geq\max_{1\leq i\leq n}|X_i|)\geq\dfrac{1}{2}$

Let $\{X_k\}_{k=1}^n$ be iid random variables that are symmetric around $0$ i.e. $X=-X$ in distribution. Define $S_n=\sum_{i=1}^nX_i$. Then show $P(|S_n|\geq\max_{1\leq i\leq n}|X_i|)\geq\dfrac{1}{2}$....
1
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1answer
167 views

Show that $S_n$ converges almost surely to $\infty$

Suppose $X_n$ are independent random variables with $P(X_n=-n^2)=\dfrac{1}{n^2}$, $P(X_n=-n^3)=\dfrac{1}{n^3}$ and $P(X_n=2)=1-\dfrac{1}{n^2}-\dfrac{1}{n^3}$ for all $n\geq2$. Let $S_n=\sum_{i=1}^nX_i$...
1
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1answer
67 views

Question about Poisson process and arrival times

Problem: On any given day you receive mail in mailbox with probability $p$. Assume whether mail is put in the mailbox or not is independent each day. If the neighbor receive mail in his mailbox with ...
4
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2answers
117 views

Proof that $\dim\text{span}\{x,x^2\cos x,\cos x\}=3$

I have the following question : Let $V$ a space of functions from $\mathbb{R}$ to $\mathbb{R}$. Proof that $\dim \text{span}\{x,x^2\cos x,\cos x\}=3$. For some reason I managed to show the opposite ...
3
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2answers
129 views

Stochastic variables independent given Tau

Say we have a filtration $(\mathbb{F}_s)$, and a stopping time $\tau$ w.r.t. to that filtration.Let $X_t$ be a continuous stochastic process (not required to be adapted to the mentioned filtration), ...
10
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1answer
369 views

The theory in probability

Consider a real-life experiment (perhaps written as a problem in a textbook): A coin is continually tossed until two consecutive heads are observed. Assume that the results of the tosses are mutually ...
2
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1answer
101 views

Independence of a hitting time and the underlying stochastic process

While I was playing around with the Girsanov's Theorem I stumbled upon the following absurdity and I couldn't resolve it with the current knowledge of stochastic analysis that I have. $B$ being ...
1
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2answers
52 views

Independence of drawing a labelled colored ball

A bag contains $5$ red balls and $5$ blue balls. The red balls are labelled $1\cdots5$ and the blue balls are labelled similarly. Let $\text{A}$ denote the event "Ball drawn is red", and $\text{B}$ ...
3
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0answers
85 views

independence of x/y and y given that x and y are not independent [closed]

Suppose two non-negative random variates x and y are NOT independent (in my case that I am interested in, the range of x is constrained by y, i.e., $0<x\leq y$). In more specific, assuming $y$ ...
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1answer
216 views

How to solve this specific Probability given 2 independent events

In a zoo, there are 1 billion monkeys. Probability that a monkey has seen a banyan tree is 0.6. Prob that monkey has seen a mango tree is 0.65. What is the minimum percentage of ...
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1answer
51 views

About Independency of Events

Is it necessary that whenever we have $\text{P(A)}\cdot \text{P(B)}=\text{P(A}\cap\text{B)}$, the events $\text{A}$ and $\text{B}$ are independent? If so, why is it a necessary condition?
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4answers
2k views

checking if matrix columns are linearly independent

according to the definition of linear dependency vectors $v_1,...,v_n$ are linearly independent $iff$ $c_1v_1$+$c_2v_2$+$...$+$c_nv_n≠0$. One can also do the gaussian elemination to get which columns ...
2
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2answers
223 views

Why are these two sigma algebras independent?

Given a probability space, let $E$ be an event and let $\{E_n\}_{n=1}^\infty $ be a sequence of events. Claim: If $\sigma(E)$ is independent of $\sigma(E_n)$ for each $n$, then $\sigma(E)$ is ...
1
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1answer
61 views

Can the independence of random variables hold for their functions?

Suppose $X$ and $Y$ are two independent continuous random variables on $\mathbb{R}$. Define: $f:\mathbb{R}\mapsto\mathbb{R}$ as a $C^\infty$ map on $\mathbb{R}$. Then is it possible to find the ...
0
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1answer
151 views

Let $A$ be a random matrix with i.i.d entries, what can we say about $Ax$?

Assume $A$ is an $m\times n$ random matrix with i.i.d entries, and $x\in\mathbb{R}^n$ be a fixed vector with $\Vert x\Vert_2=1$. Then can we say something about $y:=Ax$? Does $y$ still have i.i.d ...
0
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2answers
179 views

Statistical Dependency Transitivity

I came across this question here on Stack Exchange, and it didn't address something that I then became curious about. If $X_1, X_2$ are dependent and $X_2, X_3$ are dependent, then are $X_1, X_3$ ...
2
votes
1answer
55 views

Determine $P(S_n\leq1)$ where $S_n=\sum_{k=1}^nX_k$

Suppose that $X_n$ are i.i.d. $Uniform(0,1)$ random variables. Let $S_n=\sum_{k=1}^nX_k$ with $S_0:=0$. Then, determine $P(S_n\leq1)$. I know that maybe by using Characteristic function of $S_n$ I ...
0
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1answer
220 views

Prove that if events $A,B$ independent of C then $P(A\cap B\cap C)= P(A\cap B)P(C)$

I am trying to prove why the intersection of two events $A, B$ that are independent of C is also independent of C so that the following equality holds: $$P(A\cap B\cap C)= P(A\cap B)P(C)$$ ...
0
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1answer
55 views

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 $A$, ...
1
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1answer
68 views

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 $n$ ...
0
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1answer
709 views

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 ...
0
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1answer
131 views

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 ...
1
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1answer
49 views

Probability involving a moment generating function

Suppose that X1 and X2 are independent and identically distributed discrete random variables. The moment generating function of X1 + X2 is: M(t)= 0.01e^(-2t) + 0.15e^(-t) +0.5925 + 0.225e^(t) + 0....
2
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2answers
89 views

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 ...