Questions tagged [independence]

Question 1

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\...

Question 2

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 ...

Question 3

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} - ...

Question 4

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 ...

Question 5

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 $$...

Question 6

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 $(a,b \in \{0,1\})$, $ ...

Question 7

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=\{...

Question 8

Peano Arithmetic ($\sf PA$) differs from Robinson Arithmetic ($\sf Q$) only in that the former includes the axiom schema of Induction: $\forall x((\phi(0) \land \forall y (A(y) \to A(s(y)))) \to A(x))$...

Question 9

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 ...

Question 10

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 ...

Question 11

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 ...

Question 12

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 ...

Question 13

(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 ...

Question 14

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 $\mathcal{F}$?

Question 15

$\newcommand{\indep}{\perp \!\!\! \perp}$ Let $\{A_i\}_{i\in I}$ be finitely many events in some measure space. Is (mutual) independence of all $A_i$ equivalent to each $A_i$ being independent of all ...

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Questions tagged [independence]

Equality related to expected value and Rademacher random variable.

$Y_n$ is Cauchy distributed if $S_n/n \sim Y_1$ and $Y_i$ are symmetric [duplicate]

Proof of Poisson process property with random time

The number of positive outcomes is independent of the number of negative outcomes (under Poisson)

Does decomposition of PDFs guarantee independence of random variables?

Question on a simplish independence problem

Independence of Experiments in Conditional Coin Toss Experiments

What happens to $\sf PA$ if we remove the schema of Induction and replace it with a rule version?

some confusions about the mutual independence of the constant coefficients in the general solution of ordinary differential equations.

Random walk: Dependent or independent events when calculating "first time visit" probability

Blackwell-Girshick equation with modified assumptions

If $v_{i}$ is identically and independently distributed, will a function of all $v_{i}$ identically and independently distributed?

Paradox in the Independence of Coin Flips (Zach Star Video)

Independent Gaussian variables are still independent conditioned on a sigma algebra? [closed]

Is mutual independence equivalent to pairwise independence from finite products?

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