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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Establishing independence between two random variables

I am currently working through some basic exercises in probability and have run into a snag. I am given two independent random variables $X$ and $Y$ that are both exponentially distributed with ...
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3answers
70 views

Show that the random variables $X$ and $Y$ are uncorrelated but not independent

Show that the random variables $X$ and $Y$ are uncorrelated but not independent The given joint density is $f(x,y)=1\;\; \text{for } \; -y<x<y \; \text{and } 0<y<1$, otherwise ...
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1answer
29 views

Does independence of two random variables imply uncorrelatedness?

There are many materials about the reverse question: "Does uncorrelatedness tell us something about independence?" But how to answer the question I've posed and why? Is there some simple ...
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0answers
26 views

Finding Expectation of a probability density function Z = Y - X problem from MIT 6.041?

Problem is here 2(c) Solution is here 2(c) $$\quad \ Z\ =\ Y\ -\ X\quad\quad; 0<Y<2x$$ $$\quad\quad\ => Z\ =\ Y\ -\ X\quad\quad; -x<Y\ -\ X<x$$ $$=> Z\ =\ Y\ -\ X\quad\quad; -x<Z&...
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Show independence of $\tilde{G} : \Omega \rightarrow \mathcal{C}(\tilde{D}, \mathbb{R})$ from $\tilde{\Pi}: \Omega \rightarrow \tilde{D}$

Question Define \begin{equation} \tilde{G} : \Omega \rightarrow \mathcal{C}(\tilde{D}, \mathbb{R}), \quad \tilde{G}(\omega)(\pi) = \tilde{g}(\omega, \pi). \end{equation} How could I show, that $\...
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1answer
29 views

Dependence and correlation coefficient

Does dependence of two random variables inform us about their correlation coefficient? If so, why?
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2answers
40 views

$2 \times 2$ matrices over {$0,1$} - linearly independent

I am a student in computer science - first year. I study linear algebra $2$ - course of linear algebra $1$ . In some institutions academic studies teach the courses together / teach in another way. ...
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23 views

Is $X\perp Y$ under $\mathbb{P} (\; \cdot\; | \max\{X,Y\} \leq Z)$ if $X,Y,Z$ are independent?

Given $X,Y,Z$ are independent random variables, are $X, Y$ conditional independent with $\mathbb{P}(\;\cdot\; |X\vee Y \leq Z)$? Suppose $\mathbb{P} (X\vee Y \leq Z) > 0$, then by definition, $$\...
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0answers
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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, ...
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1answer
17 views

Is Independence Stable under Intersections?

Let $A,B,C$ be events. If $A$ and $C$ are independent, and $B$ and $C$ are independent, does it then hold that $A\cap B$ is independent of $C$ as well?
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1answer
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A Probability problem of a three-sided die with faces numbered 1, 2, and 3 from MIT 6.041

Problem is here 2(d) Original Solution is here 2(d) My approach: Let A be the event that at least one roll results in a 3 $$P(A)=1−P(no\ rolls\ resulted\ in\ 3)=1− (3/4)^6$$ Now let K be the random ...
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1answer
28 views

Independence of Solution of SDE $S^{(x_0, \sigma, \mu)}_t$ of Initial Information $\mathcal{G}_0$

Question Consider the following stochastic differential equation, given as an equivalent stochastic integral equation, where the multidimensional integrals are to be read componentwise: \begin{...
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0answers
16 views

Substituting Value for Independent Random Variable in Conditional Expectation [duplicate]

Let $\tilde{D}$ be a topological space with Borel $\sigma$-algebra $\mathcal{B}(\tilde{D})$, $\tilde{g}: \Omega \times \tilde{D} \rightarrow \mathbb{R}$ be a bounded $(\mathcal{G} \otimes \mathcal{B}(\...
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1answer
36 views

A Question related to probability.

Question is described here last question G1† Solution is here last solution Please explain the solution in simple language. And please explain how to do it using partition. My approach using ...
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2answers
37 views

Expected value of random variable ($X$) that takes non-negative integer values.

We have to proof: $$E[X] = \sum_{k=1}^{\infty} P(X \geq k).$$ We knew that: $$P(X \geq k)= \sum_{i=k}^{\infty} p_X(i)$$ $$\sum_{k=1}^{\infty} P(X \geq k)= \sum_{k=1}^{\infty} \sum_{i=k}^{\infty} ...
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0answers
35 views

Independent under regular conditional probability

I have a question on independence in probability theory. Let $X,Y$, and $Z$ are $\mathbb{R}$-valued random variables on a probability space $(\Omega,\mathcal{F},P)$. We denote by $\mathcal{F}_X$, $\...
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3answers
66 views

Linear independence of an infinite set of functions

I came across this question in one of the linear algebra textbooks: Proof the linear independence of the following set $\{f_i ∣ i\in\mathbb{N}\}$, such that $f_i : \mathbb{N}\to\mathbb{Q}$ defined as ...
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1answer
45 views

Basu's theorem to show independence

Using Basu's theorem, prove that $\sum\limits_{i = 1 }^n {(X_i - X_{(1)}) }$ and $X_{(1)}$ are independent for any $(\theta, \lambda)$. You may assume that $X_{(1)}$ is complete and sufficient for $θ$ ...
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0answers
37 views

Is this set of functions linear independent?

how do we prove that an infinite set of functions given for example as: $g \cup f_i, \text{ where } g(n)=1, f_i(i)=1 \text{ and } f_i(n)=0 \text{ for all } i \neq n, \forall i,n \in \mathbb {N}$ is ...
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2answers
48 views

How to prove linear independence of a function, if the function has different results?

I have to prove that the following set of functions $ \{ f_i \mid i \in \ \mathbb{N}\}$ is linear independent. The function is defined as followed: $f_i \in \textrm{Map}(\mathbb{N},\mathbb{Q})$ $f_i(...
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5answers
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Probability of even number of events occuring

As part of trying to fresh up on my basic probability theory I came along Ex. 1.46 in Grimmet's probability book with the second part troubling me. If $A_1$, $A_2$ , . . . , $A_m$ are independent and ...
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0answers
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Sample two individuals and find the probability of the following events

Blood can be classified according to ABO-type: $A$, $B$, $AB$ and $O$, but also according to Rh-type, $P$ (positive) and $N$ (negative). Suppose that every individual has one Rh-type and one ABO-type ...
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1answer
13 views

Independency Preseving of two Independent random variables

Suppose we have two independent random varaibles $X_1$ and $X_2$. And we have a function $a(\cdot)$. Are the two new random variables $a(X_1)$ and $a(X_2)$ still independent? For example, $a(X)=X-3$ ...
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2answers
25 views

Show that $x, cosx,$ and $\frac{x^2}{1+x^2}$ are linearly independent in $C(\mathbb{R})$?

I assume by $C(\mathbb{R})$ the question means the vector space of continuous real functions, but I'm not completely sure of that. How might I go about formally proving this? Obviously I could say ...
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3answers
132 views

Is it possible for an event $A$ to be independent from event $B$, but not the other way around?

I was wondering if event $A$ is independent of event $B$, would $B$ also be independent of event $A$? My original thought was that it should be independent, but then I realized if $A$ is independent ...
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1answer
46 views

Differential Equations- Reduction of order [duplicate]

Why is the equation in the red rectangle true? Why is it that if I have one solution y1(x), the second solution can be written as y2(x)=v(x)*y1(x)?Is it because they are linearly independent? Someone ...
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1answer
11 views

Existence of a sequence of independent $E$-valued random variables with distribution $\mu$ given $\mu$ and $E$ Polish

I know that the following question is true for $E=\mathbb{R}$. I would like to know if it can be extended to Polish spaces. Suppose that $(E,d)$ is a Polish space. Write $\mathcal{B}(E)$ for the ...
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2answers
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Linear transform of bivariate normal distribution

Suppose that $Y_1$ and $Y_2$ follow a bivariate normal distribution with parameters $\mu(Y_1)= \mu(Y_2)= 0, {\sigma^2}(Y_1)= 1, {\sigma^2}(Y_2)= 2$, and $\rho = 1/\sqrt 2$. Find a linear ...
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1answer
18 views

Questions regarding mutual independence of events [closed]

Have a few questions regarding mutual independence: If I have a set of events $A_1, A_2, …A_n$ that are all pairwise independent, it is possible that the events may not be mutually independent? If I ...
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1answer
173 views

Difference between Bernoulli random variables

Given are $n$ independent Bernoulli random variables with parameters $p_1,\dots,p_n$. We want to split them into two parts so as to minimize the expectation $\mathbb{E}[|X-Y|]$, where $X$ is the sum ...
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1answer
21 views

Product of distribution of independent random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables defined in some probability space $(\Omega,\mathcal F,P)$. Suppose that $f:\mathbb{R}^n\to\mathbb R$ is Borel measurable. I think that ...
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1answer
41 views

Are four random variables independent if pairs and sums are?

Consider four random variables $X_1, X_2, X_3, X_4$. Let $X_1$ and $X_2$ be independent as well as $X_3$ and $X_4$. We also have that $X_1+X_2$ and $X_3 + X_4$ are independent. Does this imply that ...
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0answers
26 views

Independent random variables, sigma

Let $(X_n) $ are independent integerable random variables about the same distribution. Let $S_n=X_1 +.....+X_n$ and $G_n=\sigma(S_n,S_{n+1},.....)$. Calculate : $E(X_1|G_n)$. I tried but i don't have ...
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1answer
19 views

How would one go about proving that two random variables are uncorrelated but not independent? [duplicate]

Suppose $\theta$ is a Uniform random variable on [0, 2$\pi$]. Let $X$ be $cos(\theta)$ and $Y$ be $sin(\theta)$. We have to show that $X$ and $Y$ are uncorrelated but not independent. My solution: $...
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0answers
44 views

Same birthday for two people with and without replacement

It is known that for choosing 2 people out of n with birthdays A,B the probability that A=B is $\frac{1}{365}$. In this case we have replacement which means that we may choose the same person twice. A ...
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1answer
25 views

In biased coin,What is the probability for occurrence of both head and tail at same time [closed]

In a biased coin probability of a head is P. In this coin, both head and tail cannot happen simultaneously since it is a dependent event. So I thought the answer is undefined. But the Probability of ...
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0answers
37 views

Sigma algebras generated by independent sets of events

I am interested in knowing wether the following statement is true of false. Let $ (\Omega, \Sigma , \mathbb{P})$ be a probability space and $\mathcal{A}, \mathcal{B} \subseteq \Sigma$ independant ...
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1answer
36 views

Brownian motion independence from stopping time

Let $X_t$ be a standard one dimensional Brownian motion. Let $T = \inf\{t : X_t \in\{ 1,-1\} \} $ and $S = \inf\{ t : X_t \in\{ 1, -3\}\}$ a) Explain why $X_T$ and $T$ are independent. ...
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0answers
27 views

How to determine if two RV are dependent?

Is it always possible to show that two RV are dependent, mathematically? Or do we sometimes in practise just give a valid reason after thinking a bit about the experiment when the Maths behind is very ...
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0answers
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If $X,Y$ are independent, can we show that $\text P\left[X\in A,X+Y\in B\right]=\int_A\text P\left[X\in{\rm d}x\right]\text P\left[Y\in B-x\right]$?

Let $E$ be a $\mathbb R$-vector space, $\mathcal E$ be a translation-invariant $\sigma$-algebra on $E$ (i.e. $B+x\in\mathcal E$ for all $x\in\mathcal E$) and $X,Y$ be $(E,\mathcal E)$-valued ...
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0answers
34 views

Dependent Sequence of Random Variables with Martingale Difference Property

I'm trying to come up with a sequence of non independent random variables $(X_n)$ adapted to a filtration $(\mathcal{F}_n)$ such that $$ \mathbb{E}(X_n|\mathcal{F}_{n-1})=0. $$ Can someone give me a ...
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1answer
21 views

Show diagonal covariance does not guarantee independence.

I was trying to show that if two variables have diagonal covariance, this does not necessarily guarantee their independence. For this, I was using an example where $x \sim U(-1,1)$ and $y=X^{2}$ to ...
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1answer
20 views

Apparent paradox on d-separation and conditioning in Bayesian Networks

Two random variables $A$ and $B$ are conditionally independent (when conditioned on a set of random variables $\mathcal{C}$), if the variables are d-separated, i.e. if all the paths from $A$ to $B$ ...
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4answers
62 views

Do $X,Z$ and $Y,Z$ have the same density if $X, Y \sim \text{Unif}(0,1)$, $X \perp \!\!\! \perp Y$, and $Z = X + Y$

Let $X, Y \sim \text{Unif}(0,1)$ $X \perp \!\!\! \perp Y$, and $Z = X + Y$ Is it true that $f_{X\mid Z=z}(s)$ and $f_{Y\mid Z=z}(s)$ for all $s$ (i.e. they have the same density)? Disclosure: this ...
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0answers
27 views

Which probability is affected by dependent events- P(A) or P(A|C)?

Def1. INDEPENDENT EVENTS: two events A and B are independent if occurrence of B does not affect the probability of occurrence of A (and vice versa). [A-ind-B in short] Def2. DEPENDENT EVENTS: two ...
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1answer
56 views

Prove independence of two random variables. [duplicate]

Let two random variables $\xi_1$ and $\xi_2$ be given. They are independent and have a standard normal distribution. Proove that $\frac{\xi_1^2 - \xi_2^2}{\sqrt{\xi_1^2+\xi_2^2}}$ and $\frac{2\xi_1\...
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0answers
38 views

Show that random variable are independent

Let $(X_i)_{i \geq 1}$ be independent random variables which follow the uniform law on $\{1,...,n\}$. Now we defined the random variable $\forall k \leq n, Y_k = \inf_{m \geq 1} \{ \mid \{X_1, ..., ...
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1answer
35 views

If $X_1,X_2$ are indep. and $Y_1,Y_2$ are indep. with $(Y_1,Y_2)∼N((x_1,x_2),σ^2I)$ if $(X_1,X_2)=(x_1,x_2)$, are $X_1(Y_1-X_1),X_2(Y_2-X_2)$ indep.?

Let $n\in\mathbb N$, $\sigma>0$ and $$Q(x,\;\cdot\;):=\mathcal N_n(x,\sigma^2I_n)\;\;\;\text{for }x\in\mathbb R^n.$$ Note that $Q$ is a Markov kernel on $(\mathbb R^n,\mathcal B(\mathbb R^n))$. ...
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1answer
40 views

𝑋⊥𝑌∣𝑍 and 𝑋⊥𝑍∣𝑌⇒𝑋⊥(𝑌,𝑍)⇒𝑋⊥𝑌

I'm attempting to prove this theorem from "A Concise Course in Statistical Inference" by Larry Wasserman (Theorem 17.2), picture below: . He sorta just says "this is true" and I'm left pretty ...
2
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1answer
45 views

Prove that $A, B, C$ are independent (under certain conditions)

Suppose: $A$ is independent from both $B\cap C$ and $B \cup C$. $B$ is independent from $A \cap C$. $C$ is independent from $A \cap B$. $P(A), P(B), P(C)$ are all greater than zero. Prove that $A, B,...