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.

1,445 questions
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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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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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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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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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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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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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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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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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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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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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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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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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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 $X,... 2answers 32 views E[Z=XY] not matching with Z = Binom(4,p) I tried to multiply two independent binomials, each distributed with$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)... 0answers 18 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.... 1answer 24 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 ... 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, ... 1answer 27 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 ... 1answer 40 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. 0answers 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 ... 2answers 51 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 ... 0answers 17 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 ... 0answers 10 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} $$is not i.i.d, so my understanding is that I need ... 2answers 83 views Is it true that \lim_{n \to \infty} {(P(\forall i,j\leq n \text{ } [X_i, X_j] = e))}^{\frac{1}{n}} = P(X_1 \in Z(G))? Suppose G is a group. \{X_n\}_{n = 1}^{\infty} is a sequence of i.i.d. random elements of G satisfying the condition that$$\forall H \leq G, \qquad P(X_1 \in H) = \begin{cases} \frac{1}... 1answer 60 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 ... 1answer 41 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 ... 1answer 49 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 ... 1answer 82 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 ... 2answers 180 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 ... 1answer 28 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 ... 1answer 32 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. ... 1answer 39 views Fill a joint table distribution, find covariance and check if two variable are independent. Choose numbers from {2, 3} by tossing a fair coin; the coin is tossed twice. Choose 2 if the coin turns up Heads and 3 if the coin is Tails. So the possible outcomes are {(2, 2),(2, 3),(3, 2),(3, 3)}. ... 1answer 65 views Assume that$Z_i\sim\mathcal{N}(0,1)$are independent and prove that$\sum\limits_{i=1}^{n}(Z_i-\overline{Z})^{2}\sim\chi^{2}_{(n)}$Assume that$Z_{i}\sim\mathcal{N}(0,1)$are independent and prove the following results (a)$\displaystyle\overline{Z} = \frac{1}{n}\sum_{i=1}^{n}Z_{i}\sim\mathcal{N}(0,1/n)$(b)$\overline{Z}$and$...
There are $Y_1, Y_2, \dots ,Y_n$ which are identically and independently distributed with pdf $4[(1-y)^3]$ for $0<y<1.$ We were asked to find the pdf of the first order statistic of which I got \$...