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. The tag also includes the logical property of independence in the context of mathematical logic.
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Coin Toss and Independence
In a lecture note, following is stated:
Consider the following two events. There lies in front of you a fair coin. Alice tosses it. Then Bob
tosses the same coin. Let $A$
be the event that Alice gets ...
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Equality related to expected value and Rademacher random variable.
Let $r_1,\cdots, r_N$ be real Rademacher random variables defined on a probability space $(\Omega,\Sigma,P)$, i.e., $r_1,\cdots,r_N$ are independent and $P(r=1)=P(r=-1)=1/2.$ And let $x_1,\cdots,x_N\...
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$Y_n$ is Cauchy distributed if $S_n/n \sim Y_1$ and $Y_i$ are symmetric [duplicate]
A problem from Le Gall's Measure Theory, Probability and Stochastic Processes (Chapter 9, Exercise 9.11(4)), which I'm not really sure what it is asking:
Let $(Y_n)$ be a sequence of i.i.d. real ...
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Proof of Poisson process property with random time
I have written the following statement and its proof.
Statement:
Let $\{N_t, t \geq 0\}$ be a Poisson process with rate $\lambda$ and let $T$ be a random variable independent of $N_t$ and $N_{t+h} - ...
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The number of positive outcomes is independent of the number of negative outcomes (under Poisson)
I conduct $X \sim \text{Poisson}(\lambda = 1)$ experiments. Each experiment is IID, with probability $p$ of outcome $\bf A$ and $q = 1-p$ of $\bf B$. Let $A, B$ be the total number of experiments with ...
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Does decomposition of PDFs guarantee independence of random variables?
Is this conjecture correct? If not, can it be modified to a correct one:
Let $X,Y$ be continuous RVs with joint PDF $f(x,y)$. Then $X,Y$ are independent iff there exists functions $g, h$ such that $$...
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Question on a simplish independence problem
Let $(X,Y)$ be $(\{0,1\})$-valued random variables on the same probability space. Assume $Pr[X=1,Y=1]=Pr[X=1]Pr[Y=1]$. Prove that $(X)$ and $(Y)$ are independent; i.e., for all $(a,b \in \{0,1\})$, $ ...
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Independence of Experiments in Conditional Coin Toss Experiments
Consider the following experiment: toss a fair coin until the first head appears.
The sample space is: $S=\{H_1,T_1H_2,T_1T_2H_3,T_1T_2T_3H_4,…\}$
Now, take two events: $E_1=\{T_1H_2\}$ and $E_2=\{...
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What happens to $\sf PA$ if we remove the schema of Induction and replace it with a rule version?
Peano Arithmetic ($\sf PA$) differs from Robinson Arithmetic ($\sf Q$) only in that the former includes the axiom schema of Induction:
$\forall x((\phi(0) \land \forall y (A(y) \to A(s(y)))) \to A(x))$...
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some confusions about the mutual independence of the constant coefficients in the general solution of ordinary differential equations.
I just started my undergraduate-level ordinary differential equations course. At first, the textbook used in the course provides the following definition for the solution of a general ordinary ...
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Random walk: Dependent or independent events when calculating "first time visit" probability
Let's say we have $N$ individuals $x_1,\ldots,x_N$ that move randomly on a 2D grid, all starting at the origin $(0,0)$. They have a chance of $p$ moving up/down/left/right and a chance of $1-4p$ to ...
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Blackwell-Girshick equation with modified assumptions
The question is taken from Achim Klenke's Probability Theory: A Comprehensive Course Section 5.1. There Blackwell-Girshick's equation is stated and proved with the assumption of independence of the ...
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If $v_{i}$ is identically and independently distributed, will a function of all $v_{i}$ identically and independently distributed?
A silly question for me. If $v_{i}$ is identically and independently
distributed, say normal $N\left( 0,1\right) $. Define $u_{i}=\frac{v_{i}^{2}%
}{\sum_{j=1}^{n}v_{i}^{2}},$ will $u_{i}$ be ...
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Paradox in the Independence of Coin Flips (Zach Star Video)
(Skip to last paragraph if you are famaliar the Zach Star video: https://www.youtube.com/watch?v=zczGnnM05TQ)
$P(A)=$ probability of $A$
If I flip two fair (independent) coins and tell you at least ...
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Independent Gaussian variables are still independent conditioned on a sigma algebra? [closed]
If I have two independent Gaussian random variables $X\perp Y$. Do I know that $X \perp Y|\mathcal{F}$ (independent) or at least uncorrelated, given any sigma algebra $\mathcal{F}$?
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