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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 does marginalizing over one variable affect independencies in the distribution?

I was requested to find a general algorithm which, given a Bayesian network graph $\mathcal{G}$ over a set of random variables $\mathcal{X}$ and a node to remove $X\in \mathcal{X}$, builds a new ...
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1answer
29 views

Find Mistake: Independence of two Events

Assume we have a black and a red cube with 6 sides. We definite two events A = "the black dice shows 5", B = "The product of the number of pips is a prime number". We roll the dice. So $P[A] = \frac{...
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33 views

Probability that Alice and Bob keep dating infinitely often

I solved the following problem. I would appreciate it if you can please provide feedback and let me know if I have made any mistakes. Problem statement: Online dating: On a certain day, Alice ...
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5 views

Sample Median and a single observation asymptotically independent? [on hold]

Given a sample of i.i.d. random variables $X_1,...,X_n$, can it be shown, that $X_j$ and $med\{X_1,...,X_n\}$ are asymptotically independent for any $j$? (The impact of one random variable on the ...
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1answer
24 views

One coin chosen between a biased coin and a fair coin, and is tossed n times. Find probability of having gotten the biased coin.

In a different question, I had asked for clarification on the following problem where I wanted to just understand the problem. Now, I have attempted it and wish to know if my solution is right. ...
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26 views

If three events are pairwise independent, are they independent “collectively”? [duplicate]

Three events $A, B, C$ satisfy: $P(A \cap B) = P(A) \cdot P(B)$ $P(B \cap C) = P(B) \cdot P(C)$ $P(A \cap C) = P(A) \cdot P(C)$. Does this imply $P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C)$? ...
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37 views

One of two coins chosen, tossed n times

I am looking at the following problem about a coin toss experiment. I cannot understand the statements in the problem. Problem statement is given below. A drawer contains two coins. One is an ...
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2answers
17 views

Mutual independence of three events

Is it possible to have three events $A,B,C$ such that $A$ is mutually independent to $B,C$ and $B$ is NOT mutually independent to $A,C$. By mutual independence I mean, $A$ is mutually independent to ...
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20 views

i.i.d. estimation of variance

Let ${X_i}$ for $i\in N$ be a string i.i.d. with a known expected value of $\mu = E (X_1)$. Investigate the properties the estimator of the variance given by the formula : $\frac{1}{n} \sum(X_i ^2 - \...
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1answer
56 views

Prove of $E\left(|X+Y|^a\right)\ge E\left(|Y|^a\right)$?

Let $E(V)$ be the expectation of $V$. It is also known that $E(X)=0, a>1, E\left(|X|^a\right) < +\infty$ and $E\left(|Y|^a\right)< +\infty$. $X$ and $Y$ are independent. How can I prove that $...
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2answers
24 views

On independence of collection of random variables

This question comes from the proof of Blumenthal's 0-1 law: as part of the proof, one need to show that $A$ is independent of $\sigma(B_{t_{1}},\dots,B_{t_{p}})$. The author claimed that it suffices ...
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62 views

Independent projections

Suppose, we have a matrix $X\ (n\ \times\ n) $. There are r < n independent columns(therefore r is rank).And we project our vectors with $P\ (n\ \times\ n) $ operator = $I - E$ where E is the just ...
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23 views

What makes random affine transformations pairwise independent?

Consider the affine transformation $$ h(\vec{x}) = A\vec{x} + \vec{b} \bmod p $$ for some prime $p$. Now let $\vec{x}$ be fixed and select the entries of $A$ and $\vec{b}$ uniformly at random from $\{...
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1answer
23 views

Line integral, independence of path when to use it

Let $F(x,y) = (3x^2,4y^3)$. Determine the value of $\int_c F(x,y)\cdot \mathrm dr$, where $c$ is the path from $(0,1)$ to $(\pi,-1)$ along graph of $y=\cos x$. Is it good to always check for path ...
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35 views

Suppose $A $ and $B$ are independent events. For an event $C $ such that $P(C) > 0$ , prove that the event of $A$ given $C $

Suppose $A $ and $B$ are independent events. For an event $C $ such that $P(C) > 0$ , prove that the event of $A$ given $C $ is independent of the event of $B$ given $C $ We have A and B are ...
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28 views

Minimum Independence Assumptions needed for statement to be true?

I am a bit confused on conditional independence: Given the following independence assumptions: 𝑋⫫𝑌, 𝑋⫫𝑍, 𝑌⫫𝑍, 𝑋⫫𝑌|𝑍, 𝑋⫫𝑍|𝑌,𝑌⫫𝑍|𝑋 What is the minimal set of independence assumptions ...
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13 views

The relationship between several conditional probabilities

Assume that P(A|B)=P(B|C)=t. 1) How do I calculate the intervall in which P(A|C) must lie? 2) Are there generally statable conditions (in terms of probabilistic independence for instance) when P(A|C)...
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1answer
50 views

If X, Y independent then X/Y and Y are independent?

if two random Variables $X, Y$ are independent Does that mean that $\left(\frac{\ X }{Y}\right)$ and $Y$ are independent? for example is it true that E[$\left(\frac{\ X }{Y}\right)$|$Y$]= E[$\...
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1answer
12 views

Independent random variables P(X=c)P(Y=c)=0

If $X$ and $Y$ are independent random variables and $P(X=c)P(Y=c) = 0$ for every $c$, what does it mean? Does it mean X and Y are completely two different distributions? Also I interpret it as either ...
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4answers
33 views

What is the difference between mutual independence and pairwise independence? [duplicate]

Can someone just give me a simple example and simple explanation in words about what the difference is between pairwise and mutual independence? I have this definition: Three events $A,B,C$ are ...
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1answer
33 views

Prove that $X$ and $Y$ are independent random variables.

Let $X$, $Y$ and $Z$ are random variables with density $$ f_{X,Y,Z}(x,y,z) = \frac{1}{\sqrt{\pi} \Gamma(\frac{n-1}{2}) \Gamma(\frac{n}{2})2^n}(xy-z^2)^{\frac{n-1}{2} - 1} e^{-\frac{x}{2}} e^{-\frac{y}...
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2answers
23 views

Determining whether A, B and C are independent.

Toss two fair dice. Let $A$ be the event the first die comes up odd, and let $B$ be the event the second die comes up odd. Let $C$ be the event that the sum of the values which come up is odd. Are $A,...
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1answer
77 views

Does $X - E(X|Z) \perp \!\!\! \perp Y - E(Y|Z)$ imply $(X \perp \!\!\! \perp Y) | Z$?

The question is in the title; Does $X - E(X|Z) \perp \!\!\! \perp Y - E(Y|Z)$ imply $X \perp \!\!\! \perp Y | Z$? I cant really think of a counter-example but I can't think of a proof either, any ...
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4answers
47 views

How to show $\{u_1,\sum_i^2{u_i},\cdots,\sum_i^n{u_i}\}$ is linearly independent if $\{u_1,{u_2},\cdots,{u_n}\}$?

Suppose $\{u_1,{u_2},\cdots,{u_n}\}$ is linearly independent we need to show $\{u_1,\sum_i^2{u_i},\cdots,\sum_i^n{u_i}\}$ is linearly independent. Therefore, suppose $$ \alpha_1u_1+\alpha_2\sum_i^2{...
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31 views

Independence of Intersection of Chords in a cirle

$3$ chords are uniformly selected in a circle. We need to find the distribution of number of points of intersection of chords. To solve this, I first considered an easier problem, one containing ...
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1answer
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If $\forall s\in\mathbb R^d, \forall F \in \mathcal F\ E[e^{i<s,X>}I_F]=E[e^{i<s,X>}]P[F]$ then $X$ is independen of $\mathcal F$

The claim in the title seems very plausible since the characteristic function "characterizes" or determines the distribution of $X$, but I don't know how to derive it. There is a similar result for ...
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27 views

Non-independent events

Let $\mu : \mathbb{Z}_p \rightarrow [0,1]$ be the uniform probability distribution, hence $\mu(t) = \frac{1}{p}$ for all $t \in \mathbb{Z}_p$. I am trying to find events $A,B \subset \mathbb{Z}_p$ ...
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Is $X-E[X|Y]$ independent from $Y$? [closed]

It seems this is right, but I don't know how to prove this.
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1answer
22 views

Probability of having a good set by choosing independently from Universe

Let $S$, $T$ be two disjoint subsets of a universe $U$ such that $|S| = |T| = n$. Suppose we select a random subset $R\subseteq U$ by independently sampling each element of $U$ with probability $p$; ...
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1answer
16 views

Conditionally dependent and correlated but the expected conditional covariance is zero

Recall that $$ \textrm{Cov}(X, Y | Z) = E(XY|Z) - E(X|Z)E(Y|Z) $$ and note that quite trivially we have $$ X \perp \!\!\! \perp Y | Z \Longrightarrow \textrm{Cov}(X,Y|Z) = 0 \Longrightarrow E \...
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1answer
19 views

$\{Y_i, \forall i \ge 0\}$ is iid random variable, is this equation true?

$\{Y_i, \forall i \ge 0\}$ is iid random variable, is this equation true $$ p(g(Y_{n+1}) = k_{n+1} | f_{n}(Y_0,Y_1,...,Y_n) = k_n, f_{n-1}(Y_0,Y_1,...,Y_{n-1}) = k_{n-1},...,f_0(Y_0) = k_0) = p(g(...
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21 views

Probability of max of dependent random variables

I am trying to compute the following probability: $$P(X-Y_k\leq f_k(D)-c, X-Y_i\geq f_i(D)-c\, \forall\,i: 0\leq i\leq k-1),$$ where $f_i$ are functions and $D,c$ are considered to be fixed, and $X,...
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2answers
28 views

E[Z=XY] not matching with Z = Binom(4,p)

I tried to multiply two independent binomials, each distributed with $n=2$ and probability p. So using the piecewise function for $P(XY),$ I have $$P(Z=z) = P(XY = xy) = \sum_{xy=z} z \cdot P(X=x, Y=y)...
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16 views

Proving Conditional Independence where random variables are related in a system of equations

Each of the scalar random variables, $ Y $, $ X $, $ U $, and $ V $, are continuous and possibly have $ \mathbb{R} $ as their supports. The random variable, $Z$, could be vector valued, but continuous....
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1answer
23 views

Probability calculation from distributions.

The scenario below will help me understand how to apply the law of total probability to nonbinary examples. It is inspired by an exercise from a master's program in AI using Bayesian Networks I'm ...
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1answer
23 views

Probability space and independence [closed]

I am studying probability theory and one of the questions that I have faced is this. The problem is that I either don't know where to go about with this question or even if I do do something about it, ...
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1answer
20 views

reliability of k out of n system

"a system consists of n identical component each of which is operational with probability p independent of others and a system is operational if more than half of its component working correctly, find ...
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1answer
39 views

With regard to random variables, does $(X/Y)$ independent of $(Y)$ imply that $(X)$ is independent of $(Y)$?

This makes logical sense to me, but I can't seem to prove this. Is this statement true? Note: X/Y is a ratio here, not conditioning.
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48 views

$P(a) = f(b)$, $P(b)=g(c)$; is $P(a|b,c) = P(a|b)$?

Let $A, B, C$ be random variables; $f, g$ functions; Let $A \sim f(B)$ (e.g. $A \sim N(B, \sigma)$), and $B \sim g(C)$; Question: does $P(A|B,C) = P(A|B)$ ? My intuition is that $B$ should contain ...
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2answers
42 views

Proof (not by counterexample) that $A$ and $B$ being independent given $C$ does not imply independence given $C^c$

I know that it's not true that $A$ and $B$ being independent given $C$ implies $A$ and $B$ are independent given $C^c$. However, I can only show this is false using a counterexample (see here for one ...
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16 views

Can we use the term independence for a case where something says balls are chosen simultaneously?

Consider an event of choosing a pair of balls(All the balls in bucket are distinct) from a bucket. Is it valid to think that sub-events of first ball of pair and second ball of the pair are ...
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9 views

Prove a time series to be NOT identically independent distributed

I am trying to prove that this time series (given that $X_{t}$ and $M_{t}$ are iid and independent of each other) $$ Y_{t} = X_{t}(1-X_{t-1})M_{t} $$ is not i.i.d, so my understanding is that I need ...
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73 views

Is it true that $\lim_{n \to \infty} {(P(\forall i,j\leq n \text{ } [X_i, X_j] = e))}^{\frac{1}{n}} = P(X_1 \in Z(G))$?

Suppose $G$ is a group. $\{X_n\}_{n = 1}^{\infty}$ is a sequence of i.i.d. random elements of $G$ satisfying the condition that $$\forall H \leq G, \qquad P(X_1 \in H) = \begin{cases} \frac{1}...
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1answer
46 views

$X$ and $Y$ are independent if and only if $\textrm{Cov}(f(X), g(Y)) = 0$ for all $f,g$ measurable functions?

Let $X$ and $Y$ be real-valued random variables. Does it hold that $X$ and $Y$ are independent if and only if $\textrm{Cov}(f(X), g(Y)) = 0$ for all $f,g$ measurable functions? One direction is ...
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1answer
13 views

Expressing conditional independence in terms of third conditionally independent variable

So we have three variables $A$, $B$, and $C$. We know that $A$ and $B$ are conditionally independent given $C$ (ie $P(A,B|C)=P(A|C)*P(B|C)$) How would I prove $P(C|A)=\frac{P(B|A)}{P(B|C)}$. My ...
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1answer
34 views

Conditional Independence, Decomposition

Is there some set of independence relations between three random variables $X$, $Y$ and $Z$ such that $P(Z \mid X, Y)$ = $P(Z \mid X) \cdot P(Z \mid Y)$? (I feel like there should be, but I can't find ...
3
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1answer
80 views

First orderer logic completeness and independence: the proof that disappear?

Gödel completeness theorem for the first-order logic is in fact equivalent to BPI (every proper filter can be extended to an ultrafilter). Moreover, BPI is independent of ZF; that in particular means ...
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2answers
97 views

Expected distance between leaf nodes in a binary tree

Let T be a full binary tree with $8$ leaves. (A full binary tree has every level full). Suppose that two leaves a and b of T are chosen uniformly and independently at random. The expected value of the ...
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1answer
26 views

Statistics Independence Question: Given that an item has passed inspection, what is the probability that it is actually flawed?

The problem I have a question about is below. I only have a question on part (e), but I included the other parts of the question and answers as a reference. A quality control inspector is examining ...
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1answer
25 views

Independence of increasing limits

If $\{E_n\}_{n\ge1}$, and $\{F_n\}_{n\ge1}$ are increasing and independent for each $n$, show that their limits are independent. Here is my attempt: Note that $\{E_n \cap F_n\}$ is also increasing. ...