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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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How can I prove that these events are independent?

I have a pair of sets: $A=\{n\in\mathbb{N}\mid p\cdot n\}$ $B=\{n\in\mathbb{N}\mid q\cdot n\}$ Where $p$ and $q$ are two different prime numbers. And the following event definitions: $X_n$: $n\in{...
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Help me elucidate what “the vectors $(B(d):d \in \mathcal {D_n })$ and $(Z_t:t \in \mathcal {D-D_n})$ are independent” mean.

The following is taken from a proof of Lévy's construction of Brownian motion in a book by Peter Mörters and Yuval Peres. $\mathcal {D_n } := \{\frac {k } {2^n } :1 \le k \le 2^n \} $, the set of ...
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the independence property of conditional expectation

I read a proof of following property: suppose $X$ is a random variable on the probability space $(\Omega,F,P)$, $A$ and $B$ are sub $\sigma$-field of $F$, $B$ is independent of $\sigma(X,A)$, then $E(...
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For a sequence of experiments where each $X$ is the number of trials until success with varying $p$, is each $X$ independent?

Assume that, every time you buy a box of Wheaties, you receive a picture of one of the $n$ baseball player. Let $X_k$ be the number of additional boxes you have to buy, after you have obtained $k-1$ ...
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22 views

Applying the definition of mutual independence to outcomes of random variables

My book defines mutual independence as: events ${A_1, A_2, ..A_n}$ are mutually independent if for any subset ${A_1, A_2, ..A_m}$ (where $m \leq n$) of these events we have: $$P(A_1 \cap A_2 \cap ......
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1answer
54 views

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables.

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables. My attempt: Fix $A \in \mathcal{R}$ (a Borel subset of the real ...
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1answer
24 views

Conditional Independence and product of random variables

I am stuck at the following situation: Let random variables $Y, X, W_1, W_2$. I know that $W_1$ and $W_2$ are each independent from $Y$ conditional on $X$: $$p\left(Y\mid \{X,W_1\}\right) = p\left(...
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1answer
12 views

Does time changed brownian motion have no-memory property?

Let $W=(W_t)_{t \geq 0}$ be a Browniwn motion. Do the processes $$X_t = W_{e^t} \quad \text{and} \quad Y_t = \exp \left(- \frac{t^2}{2} \right) W_{e^t}$$ have the no-memory property, i.e. are the sets ...
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Two independent geometric variables [duplicate]

Suppose that $X$ and $Y$ are independent, identically distributed, geometric random variables with parameter $p.$ Prove that $P( X = i | X + Y = n) = 1/(n-1)$ for $i = 1,2,\dots ,n-1.$ So I said $$P(...
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Conditional expectations and conditional independence

While learning causal models, I came across the below assertions: to express the conditional independence of Y and X given Z: $$ Pr(Y|X, Z) = Pr(Y|Z) $$ we can write: $$ E(Y|X, Z) = E(Y|Z) $$ If X, ...
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1answer
37 views

Expected value in a linear combination

I have a random variable Y, that is defined by: $$Y = aX_1 + bX_2$$ Where we know $X_1 $ and $X_2$ are independent. How do I write out $EX_1$ and $EX_2$ in terms of only a, b, EY, and VarY? I ...
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Proof linear dependency for vectors in $3$-D space

How to proof the linear dependency / independency ONLY using vectors (not through matrixes), as I am not familiar with this concept for now. The example is the following: Are the following vectors ...
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1answer
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About the definition of independent random vectors

I just have a question concerning the definition of independent random vectors. The random vectors $X=[X_1, \ldots, X_n]$ and $Y=[Y_1,\ldots, Y_m]$ are independent means that $X_1,...,X_n,Y_1,...,Y_m$ ...
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Independence of Events and Conditional Probability

A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with ...
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Does conditional independence of random variables transfer to functions of the random variables?

Suppose $$Y,Z \perp W |X$$ That is to say $Y$ and $Z$ are conditionally independent of $W$ given $X$. For two functions $f$ and $g$, does the following it in general hold? $$ f(Y,Z,X) \perp g(W,X) ...
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Derivation of the formula for the probability of a class, given conditionally independent attributes.

The following is a formula that finds the posterior probability of a class (i.e. yes or no) given four conditionally independent attributes: $$P(c|X) = P(x_1|c)\cdot P(x_2|c)\cdot P(x_3|c)\cdot P(x_4|...
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1answer
31 views

Why are the random variables independent?

This question is about independence in the ANOVA analysis. Assume that you have $I\cdot J$ independent random variables $X_{i,j}$. We also assume that each is normally distributed with common mean $\...
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1answer
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Patterns in probability trees

I'm just floating this and I'm not sure how to describe my problem precisely. So I'll lead with an example: Assume a team of 3 are chosen from 4 boys and 5 girls. Let the rv $X$ be the number of girls ...
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1answer
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Independent increment vs independent sigma-algebras

Suppose we want to define a Lévy process $\{ X_t \vert \ t \geq 0\} $. Is it equivalent to demand independent increments i.e. $$ \forall n \geq 1, \forall t_n \geq t_{n-1} \geq ...\geq t_1 \geq 0: X_{...
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2answers
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Probability of the intersection of two events if they are dependent

A system was to select a string uniformly at random from the set {RRRR, RS, STT} and then select a letter uniformly at random from the selected string. The system is run twice (outputting two letters)...
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2answers
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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
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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
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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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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
26 views

Dependence and correlation coefficient

Does dependence of two random variables inform us about their correlation coefficient? If so, why?
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$2 \times 2$ 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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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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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
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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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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
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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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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
35 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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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
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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
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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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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
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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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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
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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
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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
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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
42 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
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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
61 views

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
17 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
165 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 ...