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 the tags (probability), (probability-theory) or (statistics). Do not use for linear independence of vectors and such.

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Why are this two stochastic processes independent?

Let us consider a filtered probability space $(\Omega, (\mathcal{F}_t)_t, \Bbb{P})$, and let $M$ be an $((\mathcal{F}_t)_t, \Bbb{P})$-continuous martingale. Let $T_t:=\inf\{s: \langle M\rangle_s>t\}...
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Prove the existence of $g$ such that $\mathbb E (e^{aX_t} ) = e^{g(a) t}$ for a Compound Poisson Process $X_t$

I have been studying a course on Stochastic Processes and recently encountered the following result with the proof listed as an exercise. Let $X_t$ be a Compound Poisson Process defined by the sum of ...
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show that an expression $E(x,y)$ does not depend on x

the question Let $a,b,d,e$ be real numbers with $a>d$ and $c=a^2+b^2, f=d^2+e^2, m=\frac{b-e}{a-d} , n=\frac{bd-ae}{a-d}$. Show that if $x\in [-a,-d] $ and $y=mx+n$, then $$E(x,y)=\sqrt{x^2+y^2+2ax+...
IONELA BUCIU's user avatar
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Are random variables X and Y=e^X independent? [closed]

X - random variable. We define Y=e^X. Are X and Y independent?
Jan Nowak's user avatar
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If one event influences the other, are they independent?

Consider some human action, C. It could be to commit a crime like launching a cyber-attack or embezzling funds. How likely is it that there is an actor that intends to attempt C in the next year? Call ...
Tupelo Thistlehead's user avatar
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35 views

Conditional Joint distribution decomposition

Let $I$ be an index set and $\mathcal{X}_i$ for $i\in I$ be a standard measurable space (or say a Polish space or $\mathbb{R}$). Suppose that the probability measure $P(X_I)$ on the product space $\...
Mars's user avatar
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Independence of union of disjoint intersection of independent events

I have a family of mutually independent events $(B_i)_{i \in \mathbb N}$, each having the same probability. For $k \in \mathbb N$, let $A_k$ be event "$k$ consecutive $B_i$'s occur between steps $...
user8171079's user avatar
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Can anyone answer Part C and explain to me if I am doing correctly? [closed]

I'm attempting to solve Part C which I know involves comparing the product of the marginal of x and y to f(x|y). If they are the same then they are independent. I got the marginal of y as 1/(2sqrt(pi))...
Guhan Gnanam's user avatar
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'Compactness' result for independent sigma algebras

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $A, B_1, B_2, \dots \in \mathcal{F}$. Let $\mathcal{F}_n := \sigma(B_1,\dots,B_n)$ and $\mathcal{F}_\infty := \sigma(B_i : i \in \...
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independently generated vs. iid

I've seen in some texts; given a random variable $X$, independently generate a copy of $X$ denoted by $Y$. What's the difference between this and just saying "let $X,Y$ be i.i.d"? They seem ...
asuuuka's user avatar
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Independence of Hilbert-space-valued random variable

Let $X,Y$ be two independent random variables taking values in a separable Hilbert space $U$. Prove that $X$ and $Y$ are independent if and only if for all $u,v\in U$, $(X,u)$ and $(Y,v)$ are ...
George's user avatar
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Can someone explain these rules about determining linear independence in matrices that my professor taught?

So the way that I was taught to determine linear independence and come up with a dependence relationship is to get a matrix, put it into reduced row echelon form, and then make that all equal to zero. ...
AnonymousThankfulPerson's user avatar
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Proving $A_1^c,A_2, ... ,A_n$ are independent.

Let $A_1,\ldots, A_n$ be independent events. Prove $A_1^c,A_2,\ldots, A_n$ are independent as well. So I wish to reference a question posted by Jeffrey Anders: How can I prove that $A_1^c, A_2,A_3,...
MathStudent101's user avatar
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Do I need additional assumptions for this equality to hold?

Suppose I have three random variables $X_1,X_2, V$, and I want the following condition to hold: $E[X_1^2|X_2<V<X_1,X_1,X_2]=E[X_1^2|X_1]=h(X_1)$, i.e., I want conditioning varibles $V$ and $X_2$...
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Two Gaussian which are not jointly Gaussians but not dependent

Let $X \sim N(0,1)$ and B such that $P(B=1)=P(B=-1)=1/2$, two independent variables. Define - Y=BX. Then Y is also Gaussian since - $$F_{Y}(y)=P(Y\leq y)=P(BX \leq y)=P(B=1)P(BX \leq y|B=1)+P(B=-1)P(...
Yar Sha's user avatar
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Empirical Distribution Convergence, Ordering Of Samples

I am trying to formally justisty a "rearrangement" algorithm, which rearranges the samples of two random variables to reflect a certain joint distribution. Suppose that we have two pairs $(...
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2 answers
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Show that if $\mathbb{P}(A|B) = \mathbb{P}(A |\Omega \setminus B)$ and $ 0\lt \mathbb{P}(A),\mathbb{P}(B) \lt 1$ that A and B are independent events

I for some reason can't solve this and I have issues visualizing solutions that come with stochastical independence. It seems intuitive for me though, that if $\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)...
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Independence from random vector eith joint density function

If (X,Y) is a random vector with the following joint density function f(x,y) =\begin{cases} \pi^{-1} \; \; x^2 + y^2\leq 1 \\ 0 \; \text{elsewhere} \end{cases} Are X and Y independent? ...
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Conditional probability of an intersection of 3 independent events

We are given the events $A$, $B$, $C$ and I, where $A$, $B$, $C$ are independent. We know the following probabilities: $P(I)$, $P(A|I)$, $P(B|I)$, $P(C|I)$, $P(A|\overline I)$, $P(B|\overline I)$, $P(...
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If $f(X)$ and $g(Y)$ are independent for all measurable $f, g$, then $X, Y$ are independent.

In Jean Jacod's Probability Essentials Theorem 10.1, the proof for if $f(X), g(Y)$ are independent for all measurable functions $f, g$, then the random variables $X, Y$ are independent chose ...
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Expectation of function of dependent random variables

Consider a real valued function $h$ on random variables, and the following sample-average function $h_S(X_1, X_2, ..., X_n) = \frac{1}{n}\sum_{i=1}^n h(X_i).$ Assume $X_1, X_2, ..., X_n$ are not ...
tempered_overfit's user avatar
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1 answer
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$X_1,....,X_n$ i.i.d. random variables $\sim \mathcal{N}(\mu, \sigma^2 )$, $Y = (X_1-\bar{X}, ..., X_n -\bar{X})$, find MGF of Y.

The question: Let $X_1,....,X_n$ i.i.d. random variables $\sim N(\mu, \sigma^2 )$, and $Y = (X_1-\bar{X}, ..., X_n -\bar{X})$, show that $$M_Y(t) = \exp \left\{ \frac{\sigma^2}{2} \sum_{i=1}^n (t_i - \...
The Math Hermit's user avatar
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114 views

Conditional expectation and independence with respect to $\sigma$-algebra and null set

This question concerns an extension of a previously solved question regarding measure-theoretic definitions of conditional expectation and conditional independence with respect to a $\sigma$-algebra ...
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Proving $\mathrm E[XY] = E[X] E[Y]$ whenever $X$ and $Y$ are independent random variables.

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability measure space. Let $X$ and $Y$ be two independent random variables on it. Then $\mathrm E [XY] = \mathrm E [X] \mathrm E [Y].$ I tried to do it ...
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Can an IID sequence be a martingale?

I was looking into Doob's upcrossing inequality, which says that for a supermartingale $X$ and real numbers $b > a$, $$(b-a)\mathbb{E}[N_n([a,b], X)] \leq \mathbb{E}[(X_n -a)^-],$$ where $N_n([a,b],...
Harry Partridge's user avatar
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2 answers
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Are the probabilities for a run in a coin toss experiment starting at a certain index independent?

I am asking myself the following question: Consider $n$ i.i.d. throws of a fair coin and let $I_{k,i}$ be the indicator RV for the event that a run of $k$ consecutive heads starts at the $i$-th throw....
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1 vote
2 answers
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Why is the event $\left \{\omega \in \Omega\ \bigg |\ \sum\limits_{n \geq 1} X_n (\omega) \lt \infty \right \}$ considered to be a tail event?

Let $X_1, X_2, \cdots$ be a sequence of independent random variables in a measure space $(\Omega, \mathcal F, P).$ Then why is the event $\left \{\omega \in \Omega\ \bigg |\ \sum\limits_{n \geq 1} X_n ...
Anacardium's user avatar
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1 vote
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Question on independence of random variables.

I have a question on independence of random variables. Question $:$ Let $X_1,X_2, \cdots, X_n$ be independent random variables. Let $A = \{i_1, i_2, \cdots, i_p \}$ and $B = \{j_1, j_2, \cdots, j_q \}$...
Anacardium's user avatar
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1 vote
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Expected value of absolute value of sum of independent variables (Convergence in probability)

Given $X_1, X_2,..., X_n$ independent random variables with $P(X_n=k^n)=P(X_n=-k^n)=1/2$ (assuming $k$ is an arbitrary constant). Let $S_n = X_1 + X_2 +... + X_n$. Determine for which $\epsilon>0$, ...
tt99999's user avatar
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2 answers
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Does "not mean independent" imply "not independent"?

If X and Y are two random variables such that $E(X | Y = y) \neq E (X)$ for some $y \in D_y$, then X is not mean independent from Y. Does it imply that X and Y are also not stochastically independent?
Pedro's user avatar
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1 answer
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Trouble with showing conditional independence in a channel

In pg. 193 of Cover's Information Theory textbook, a discrete memoryless channel (DMC) is defined by the triplet $(\mathcal{X}, p(y \mid x), \mathcal{Y})$, and its nth extension is defined as the ...
user1143399's user avatar
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A question about sample events and binomial probability

!Disclaimer: Since English is my second language, I apologize beforehand if my writings aren't that coherent. So I'm having some problems understanding the question of this particular problem, ...
rutkre's user avatar
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0 answers
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gamma distribution independent variables joint density and transformations

The probability density function of gamma distribution is given like $f(x)=\begin{cases}\frac{\beta^{a}x^{a-1}e^{-\beta x}}{\Gamma({a})}& \text{x>0}\\ 0,\text{elsewhere}\end{cases}$ Let X and Y ...
maths and chess's user avatar
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Why is $x_1$ independent of $x_2$ for a joint gaussian with $0$ correlation

I am reading Murphys machine learning: a probabilistic perspective and in 4.3.2.1 it says that for a joint gaussian with 2 zero mean random variables, lets say $y$ and $x$, if the correlation is $0$, ...
NicholasCostanza's user avatar
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Minimize the joint entropy of Bernoulli variables with given marginal distributions

Given a list of Bernoulli variables $X_1, X_2, \ldots, X_n$ with fixed marginal distributions (i.e., success probabilities) $p_i = \Pr[X_i = 1], \forall i \in [n]$. Question: How can we minimize the ...
Vezen BU's user avatar
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Is conditional independence of random variables preserved under diffeomorphisms?

I have a general question to understand when conditional independence is preserved. Let $(X, Y, \mathbf{Z})$ where $X$ and $Y$ are random variables and $\mathbf{Z}$ is a random vector. If $X \perp Y | ...
ajl123's user avatar
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Clarification about a little doubt on the independence definition

A simple question I tried to ask in a little different form on another site, evidently without much success in making myself clear. I am retrying with (hopefully) clearer language. I was looking at ...
Jada's user avatar
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How to measure the (in)dependence of multiple random variables?

To measure the (in)dependence between two random variables, we can use correlation coefficients. See, e.g., How to measure the independence of random variables?. What about multiple random variables? ...
Vezen BU's user avatar
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Help with Understanding Independent Events

This is a question from MIT 6.041 open courseware. Most mornings, Victor checks the weather report before deciding whether to carry an umbrella. If the forecast is “rain,” the probability of actually ...
John Cho's user avatar
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Pairwise Independence

Okay, so recently, I asked a question about two siblings who are playing $10$ rounds of a soccer/football game. James has a $70%$ chance of scoring each time and John has a $60%$ chance of scoring ...
River Uzoma's user avatar
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Independence of Variables and Covariance

I am quite confused on the concept of independence of random Variables. Everything seemed to be fine until random vectors were introduced. In our lecture notes is says $\vec{X}=(X_1,...,X_n)$ is a ...
Henrie Küppers's user avatar
2 votes
1 answer
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Convex Independence

I have a difficulty in understanding a given definition for convex independence: A set of beliefs $\beta$ of an agent $i$ satisfy convex independence if beliefs of no type $t_i$ can be represented as ...
asdf1234's user avatar
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Question regarding independent (Bernoulli) trials: rolling a die 10 times and defining success as rolling a 1 or a 6.

I just want to make sure that the scenario I'm describing is indeed a Bernoulli trial type of experiment. Here is the set up and query regarding the set up: A six-sided die is rolled 10 times. What is ...
Mariusz Popieluch's user avatar
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1 answer
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if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis.

here is the question I am trying to solve: In a matroid $M,$ if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis. I know how to prove that a set ...
Intuition's user avatar
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Marginalization of the conditional expectation

Let $Y, Z, X_1, X_2, T$ be random variables. Let $\mathbb{E}(Z\mid X_1, X_2) = \mathbb{E}[Y\mid T=t, X_1, X_2]$. I need a property $$\mathbb{E}(Z\mid X_1) = \mathbb{E}[Y\mid T=t, X_1].$$ Does it hold ...
Albert Paradek's user avatar
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30 views

Conditional density equalling the marginal does not imply independence?

If $X$ and $Y$ are independent random variables, then $f_{X|Y}(x|y) = f_X(x)$. Is the converse false? I think it's false because suppose you have a set of i.i.d. random variables, $\{X_i\}$, with ...
johnsmith's user avatar
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Question about Expectation

Is my thinking correct? Let's suppose there are two independent random complex vectors, $a,b\in\mathbb{C}^N$. Additionally, both vectors have a zero mean, and their covariance matrices are defined (e....
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In a generator matrix of a linear block code ,how does increasing linear vectors in a field $F^k$ has $q^k-q^i$ choices?

I am trying to study error control and coding theory by myself. There is an unsolved question which says that the total number of distinct generator matrices of a linear [n,k,d] code over $F=GF(q)$ is ...
Userhanu's user avatar
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Independence of Limiting Event

Let $A_1\supset A_2\supset A_3,\ldots$ and $A_n\rightarrow A$. If $B$ is independent of all $A_n$, then it must be independent of $A$ also, right? I showed this by first noting that $B\cap A_1\supset ...
Ian L's user avatar
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Basic probability independence in roulette

I am teaching my students how roulette works and we were talking about the probability of betting on the first column of numbers $\{1,4,7,10, 13, 16, 19, 22, 25, 28, 31, 34\}$ and the probability of ...
CCHS Math's user avatar

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