# 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 \$...