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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Product and sum of iid random variables converges a.s.

The set of independent and identically distributed r.v. $Y_{1},Y_{2},...,Y_{n}$ has a distribution $Pr(Y_i=1)=Pr(Y_i=1/2)=1/2\ \forall i>0$ . If $Z_n=\prod_{i=1}^{n}Y_i$ and $S_n=\sum_{i=1}^{n}Z_i \...
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A and $B \cup C$ are independent. Please give an example to show whether A and B are independent. [closed]

There are 3 events $A, B$, and $C$. Assuming A and $B \cup C$ are independent. Please give an example to show whether A and B are independent. Please give an example to show whether A and B are ...
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A question regarding conditional independence

I have the following random variables: $X= Z + V $ and $ Y $, where $ (V, Y)\perp \!\!\! \perp Z $ but $ Y \not\!\perp\!\!\!\perp V $. Is it true that $ Y \perp \!\!\! \perp X = Z+V |V $? ...
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stochastic independence

I would like to understand what does here mean the table 3.7.1. It has apparently 5 entries instead of just 4. There are 2 same pages each taken from a different source. EDIT this frame was taken ...
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Balanced splits in Randomized Quicksort

I am reading about usage of Chernoff bounds for bounding number of comparisons in a randomized Quicksort from here. What is proved is that ($n$ is number of elements in the array) $$\Pr[\text{Number ...
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I don't think W/O replacement events are dependent.

Let say there're cards numbered 1 to 10. P(A) is the probability you pick a card whose number is the multiple of 2. P(B) is the probability you pick a card whose number is the multiple of 5. So now ...
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An example about independence between three random variables. [closed]

Say I have three random variables: $A$, $B$ and $C$. Now say that $A$ and $B$ are independent, $B$ and $C$ are independent, and $A$ and $C$ are independent. I need a practical example (i.e., a ...
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Showing if two events determined by a die toss are independent or not.

I'm having trouble setting up a problem with random variables to determine if two events A and B are independent or not. The problem is as follows: A fair die is repeatedly toss until the first even ...
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$A_1,…A_n\;$ are independent events iff $A_1,…A_{n-1},A_{n}^{c}\;$ are independent events

I know that if $A_1,...A_n\;$ are independent events, then we have forall $I\subseteq[n]\;$ $P(\bigcap_{i\in I}A_{i})=\prod_{i=1}^{n}P(A_{i})$ I am kind of stuck, I guess I should be using de morgans ...
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If $X, A_1$ are independent, and $X, A_2$ are independent, then are $X, A_1 \cdot A_2$ independent?

There are RVs $A_1, ..., A_n$ and $X$. $A_1$ and $X$ are independent, $A_2$ and $X$ are independent, ... $A_n$ and $X$ are independent. Does it mean that $A_1 \cdot ... \cdot A_n$ and $X$ are also ...
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Independence of marginals and conditionals

Given a joint distribution over random variables (E,C), is there a sensible way to understand what it would mean / how to prove if (E|C) and C were independent?
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Maximum Likelihood and Sampling/Independence

If I viewed the data of a unknown weighted coin that was flipped N times and I see that there are H heads and T tails, I know that the maximum likelihood estimation for the probability of heads is ...
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Does CDF have same conditional independence structure as PDF?

Conditional independence structure of random variables determines how the joint p.d.f factorizes. Does the corresponding (cumulative) distribution function exhibit the same conditional independence ...
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Prove that the number of vectors in LI subset can not exceed the number of vectors in a generating set of vector space?

Any subset 'T' of vector space is the generating set of vector space if every elements of vector space can be expressed as the linear combination of elements of 'T'. Let S be the LI subset of vector ...
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Does uniformly bounded second moment imply Lindeberg's condition>

Suppose an independent sequence $X_i \in \mathbb{R}^k$ is such that $\sup_n E(\Vert X_i \Vert^2) \leq M$ for some constant $M$. Does this imply that $X_n$ satisfies the following Lindeberg-type ...
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Conditional probability and independent sigma algebras

Exercise: Given a random variable $X$ , sigma algebras $\mathcal{G},\mathcal{H}$ such that $\mathcal{H}$ is independent from both $\mathcal{G}$ and $X$ prove that $\operatorname{E}[X\mid\sigma(\...
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Proving that $P(B|C \cap A^{c}) = P(B|C)$

I was solving the following question: Most mornings, Victor checks the weather report before deciding whether to carry an umbrella. If the forecast is “rain,” the probability of actually having rain ...
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Concentration for product of independent random variables

I have two sequences of random variables: $X_1, \ldots, X_n$ are iid copies of a bounded zero-mean random variable X, and $Y_1, \ldots, Y_n$ are iid copies of a zero-mean random variable $Y$. $X$ and $...
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Identical distributions

I am wondering if the following holds/can be proven: Let $X_1, \ldots, X_n$ be iid copies of a random variable $X$, and let $Y_1, \ldots, Y_n$ be iid copies of a random variable $Y$. $X$ and $Y$ are ...
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Independence of Joint and Marginal Distributions

If I know two joint distributions are independent, say $f_{XY}(x, y)$ and $f_{VW}(v,w)$, then could I then say that marginally, $X$ and $V$ or $X$ and $W$ are independent? Also, I assume that just on ...
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Stochastic process, modification independence

Let us have an i.i.d. process, say $(X_n)_{n \in N}$ and a continuous modification $(Y_n)_{n \in N}$. I need to show that also $(Y_n)_{n \in N}$ is an independent process. I started considering the ...
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Properties of Expectations with Joint Distributions and Independence

Suppose I have a random vector $(X, Y)$ governed by some $f_{XY}(x,y)$, and another random vector $(A, B)$ with the exact same joint distribution of $(X,Y)$, but $(A, B)$ is independent of $(X,Y)$. If ...
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Is independence inherited by modifications when the original process is iid

Consider a sequence of iid random variables $(X_{n})_{n\in\mathbb N}$ as well as a modification of the original sequence, namely $(Y_{n})_{n \in \mathbb N}$. Is independence inherited by $(Y_{n})_{n \...
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Independence Check for Music Singing

The dataset is constructed by 1st interval, 2nd interval......to 24th interval. each interval is measured by the error between targeted cents and people actual singing. For example, if the 1st ...
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How does the independent events rule prove independence?

We are taught to find if events A and B are independent it will satisfy the equation P(A and B) = P(A)P(B), but how does this prove independence.
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Sample variance and independence

If I have a random sample (X1,...Xn) from one normal population, and another random sample (Y1,...Yn) from another normal population where these samples are independent of each other. Is it true that ...
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Pairwise independence of arbitrary product of random variables

I have the following problem: Let $X_1, \ldots , X_n$ independent random variables such that, $\forall \ i \in \{1,2,\ldots, n\}$ : $$ \mathbb{P}[X_i = 1] = \dfrac{1}{2} = \mathbb{P}[X_i = -1] $$ Let $...
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I literally can't understand the Independence Probability

They say the coin flipping are independent events each other. (Gambler's fallacy). And they also say that a set of probabilities is independent each other, if they satisfy P(A∩B) = P(B│A)×P(A) = P(A)×...
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How can I find set variables by which two nodes in a causal graph are d-separated by?

For example, in this graph Graph G how would I be able to determine what variables allow X1 and X4 to be d-separated. Would it be X6 and X7 because they have no connection to X1 and X4? What is the ...
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Independent random variables and Expectation

So, I know that if two random variables X and Y are independent, then E(X|Y) = E(X). And I read that the inverse of this statement may not be necessarily true. ie. E(A|B) = E(A) but A and B are not ...
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Independence of two coin tosses

Two fair coins are tossed (independently of each other). Let A be the event that both coins show head, and B be the event that both coins show tail. Are A and B independent? I'm a little confused ...
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“ A 7 is drawn from a deck of cards, then without replacing the card, a 2 is drawn.” is NOT an independent event?

Two events are independent if the outcome of one event does not affect the outcome of the other event. One of the following statements does NOT describe independent events. Which one? F. A coin lands ...
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Probability of a random variable is greater than its predecessors

Suppose I have a sequence of iid random variables $(X_n)_{n\in \mathbb{N}}$ would I be correct in thinking that the following is true? $$\mathbb{P}(X_k>X_1,\cdots, X_{k-1})=\mathbb{P}(X_k>X_1)^{...
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Clarifying an identity for random variables

In this note on the laws of large numbers, on page 4 the author comes to a sum of terms of the forms $E[X_i X_j X_k X_\ell], E[X_i^2 X_j X_k], E[X_i^2 X_j^2], E[X_i^3 X_k], E[X_i^4]$, where $X_i, X_j, ...
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Paradox with deterministic colliders in directed acyclical graphs

I encounter a paradox with a collider in a DAG (directed acyclical graph) and would appreciate some guidance. In this case the collider $Y$ is a deterministic function of variable $X$ and a second set ...
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“Gram-Schmidt”-like decomposition method for creating independent variables.

Considering $X$ and $Y$ two variables "unlinked", i.e there is no deterministic $f$ function so that $Y = f(X)$, is there any theorem ensuring existence/characterization of some $g, h$ ...
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If concentration inequalities correspond to non-asymptotic probability, what do anti-concentration inequalities correspond to?

Concentration of measures is a niche in statistical probability theory, most notable for deriving concentration inequalities. I'm still trying to grasp its underlying motivation and premise compared ...
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Proof of the product rule in probability theory for causal independence

There is a common attitude in the text books on probability that the so-called product rule is an obvious property, when events are independent, i.e., $P(A\cap B)=P(A)P(B)$ when $A$ and $B$ are ...
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A normal random variable independent of each component of a multivariate normal random vector.

Suppose that $\mathbf{Y}\sim N_3(0,\,\sigma^2\mathbf{I}_3)$ and that $Y_0$ is $N(0,\,\sigma_0^2)$, independently of the $Y_i$'s. My question is that does $(\mathbf{Y}, Y_0)$ also have multivariate ...
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Variables with 0 Covariance that are NOT independent.

I have a few examples I could provide, here is one. Let X = Uniform(-1,1) and X = $Y^2$. Here we have clearly E(X) = 0 and then E(Y) = $\frac{1}{3}$. Cov(X,Y) = E(X*Y) - E(X) * E(Y) = E($X^3$) - 0 = E(...
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Independence/conditional probabilities

Does $P(A|B,C) = P(A|C) => P(A,B) = P(A)*P(B)?$ And if so, how do I prove it? I would intuitively say this is true since $P(A|B) = P(A)$ would mean $A$ and $B$ are independent and hence $P(A,B) = ...
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What is the probability that those events are independent?

Suppose during some experiment two events $A$ and $B$ may occur. Before the experimentation started, nobody had any knowledge either about whether those events are independent or anything about their ...
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If $F_1$ and $F_2$ are both independent of $F_3$ and independent of each other, is $\sigma(F_1\cup F_2)$ independent of $F_3$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\mathcal F_i\subseteq\mathcal A$. Remember that $\mathcal F_1$ and $\mathcal F_2$ are called ($\operatorname P$)-independent if $$\...
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Equivalent condition for a process to have independent increments

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$, $E$ be a $\mathbb R$-Banach space and $(X_t)_{t\ge0}$ be an $E$-...
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Linear dependency in $\mathbb{R}^4$

Show that the three vectors are linearly dependent in $\mathbb{R}^4$. $u=(0,3,1,-1)\ v=(6,0,5,1)\ w=(4,-7,1,3)$ They are linearly independent if $c_{1}u+c_{2}v+c_{3}w=0\Rightarrow c_{1}=c_{2}=c_{3}=0$ ...
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If the columns of AB are linearly independent, how can I prove the columns of B must be linearly independent? [duplicate]

So a matrix, A, is linearly independent if Ax = 0 has only the trivial solution (x=0) so let A = AB, then if the columns of AB are linearly independent, the equation (AB)x = 0 has only the trivial ...
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Given that random variables $X$ and $Y$ are independent when are $X+Y$ and $X-Y$ independent?

Given that random variables $X$ and $Y$ are independent I can come up with an example where $X+Y$ and $X-Y$ are not independent e.g. $X$ and $Y$ take values either $0$ or $1$ with equal probability. ...
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If $(X_k)_{k\in\mathbb N}$, show that $\sum_{k=1}^n(X_k-\text E[X_k])$ converges almost surely as $n\to\infty$

Let $\nu$ be a $\sigma$-finite measure on $\mathbb R$ with $$\int_{(-1,\:1)}\nu({\rm d}x)x^2<\infty\tag1$$ and $\nu(\{0\})=0$, $$I_k:=\left(-\frac1k,-\frac1{k+1}\right]\cup\left[\frac1k,\frac1{k+1}\...
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Is the given discrete random vector independent?

A random vector $(X,Y)$ has PMF given by $p(1,-1) = p(-1,1) = 1/2$. Are $X,Y$ independent? I think they are dependent because $p(X=1)=p(Y= -1)= 1/2$ and $p(X=1,Y= -1)\ne p(X=1)*p(Y= -1)$. Is this ...
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Does intersection mean set theoretic point of view for independent events?

It is evident that when two events $A$ and $B$ are independent, then $P(A\cap B)=P(A)P(B)$ A good example is, "Tossing a coin and rolling a die". What is the probability that heads occurs on ...

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