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.

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14 views

If I have $(\log(L(T_{i}))_{i=1,...,n}$ are respectively $\mathcal{N}(\mu_{i},\sigma_{i}^{2})$-distributed, what about the following distribution

Consider the equidistant discretization $0=T_{0}< ...< T_{n}=T$ of $[0,T]$ . Consider the collection $(\log(L(T_{i}))_{i=1,...,n}$ such that $\log(L(T_{i}))\sim\mathcal{N}(\mu_{i},\sigma_{i}^{2})...
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72 views

Simplify Probability Expression (Cornfield Inequality)

I have an expression like $P(C|SR)$, and I know $S$ and $R$ are conditionally independent given $C$ (EDIT they are also independent given $C^{\complement}$). I want to simplify it into a certain form. ...
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66 views

Independence of random variables product of expectations proof

I want to prove that $X_1$ and $X_2$ are independent if and only if for all continuous and positive functions $f_1$ and $f_2$ $$\mathbb{E}[f_1(X_1)f_2(X_2)]=\mathbb{E}[f_1(X_1)]\mathbb{E}[f_2(X_2)]$$ ...
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30 views

Can dependency between variables increase the resulting entropy?

Let $n,d\in\mathbb N^+$ and let $\mathcal D\in\Delta(\{1,\ldots,n\})$ be a probability distribution over $\{1,\ldots,n\}$. Consider random variables $X_1,\ldots,X_d, Y_1,\ldots, Y_d\sim \mathcal D$. ...
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56 views

Writing a random variable as the sum of independent random variables

Let $X_i$ be independent random variables with $X_i \sim Po(1)$. We know (for instance by looking at characteristic functions) that then $\sum_{i=1}^{n} X_i \sim Po(n)$. I am interested in the ...
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58 views

Independence of two statistics

I was provided a question which went as follows: If $Y_1,Y_2,...,Y_n$ is a random sample from $N\left(\mu,\sigma^2\right)$, prove that $\bar{Y}$ is independent of $\sum_{i=1}^{n-1}\left(Y_i-Y_{i+1}\...
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31 views

Cramér-Rao lower bound - Estimator is independent of the parameter

I have followed the Cramér-Rao lower bound (CRLB) derivation, and I couldn't figure out why - If $f(x; \theta)$ be a probability density with continuous parameter $\theta$, and $X_1, \dots, X_n$ be ...
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24 views

Conditional Independence Given Complement of the Conditional

I am running into a problem with two tests run serially. Sensitivity Specificity Test 1 0.95 0.90 Test 2 0.94 0.89 Tests are run serially, thus both tests must pass in order for the test battery ...
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34 views

Quick question about equality of random variables

I have two RV $X$ and $Y$, and I am asked to find the probability of $P(X=Y)$. To tackle to problem, I've conditioned on $X$ and used the law of total probability to write : $ P(X = Y) = \...
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37 views

Joint cdf and conditional expectation problem

Suppose $X, Y$ are continuous RVs with joint pdf $f(x, y) = 0.5$ for $0\leq x\leq y\leq 2$ and $f(x, y) = 0$ otherwise. (i) Find the cdf of $Y$. (ii) Compute $P(X < 0.5 | Y = 1.5)$. Are $X$ and $Y$ ...
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47 views

Expected value of sin of sum of n random angles

Consider the following problem. Let $θ_1,θ_2,...θ_n ∈[0, \frac{π}{2}]$ be independent and uniformly distributed variables. Find $E[sin(θ_1 + ... + θ_n)].$ I was able to solve for $n=1$ (of course), ...
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Sequence generated by difference of Gaussian random variables.

I tried a question and intutively i can guess it's answer, but I am not sure if my logic is correct or not: Let $\{X_1,X_2, \dots ,X_n\}$ be a sequence of independent and identically distributed (i.i....
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Redundant constraints and linear dependency

Consider the following polyhedron $S=\{Ax\leq b \mid x\geq 0\}$, where $A\in \mathbb{R}^{n\times m}$ and $b\in\mathbb{R}^n$. Is the following statement true: If an inequality $A_k x\leq b_k$ for $k\in ...
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51 views

Maximum possible probability given null interesection of events

I was working on some random probability problems given in exams throughout US colleges, and came across this relatively simple problem that is giving me a bit of trouble. Suppose that $A$, $B$, and $...
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42 views

How can I show that the maximum is 0?

I am trying to solve this exercise (I don't know if it's from a book, feel welcome to credit it if you've been it before). It says that X,Y are two independent, equal random variables. If their (...
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36 views

Which of the following properties does a process with independent increments really admit?

Let $E$ be a normed $\mathbb R$-vector space and $(X_t)_{t\ge0}$ be an $E$-valued adapted process on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$ such that ...
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35 views

If $(\mathcal F_1,\mathcal F_2,\mathcal F_3)$ is independent, is $\mathcal F_1\vee\mathcal F_2$ independent of $\mathcal F_3$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\mathcal F_i\subseteq\mathcal A$ be a $\sigma$-algebra on $\Omega$. If $(\mathcal F_1,\mathcal F_2,\mathcal F_3)$ is independent, ...
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22 views

If $X_t-X_s$ is independent of $\mathcal F_s$ for all $t\ge s$, is $(X_t-X_s)_{t\ge s}$ independent of $\mathcal F_s$ as well?

Let $(X_t)_{t\ge0}$ be a process on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$ such that $X_t-X_s$ is independent of $\mathcal F_s$ for all $t\ge s\ge0$. ...
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If $X$ has increments independent of $\mathcal F$, is $\Delta X_t$ independent of $\mathcal F_s$ for all $t>s$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$, $E$ be a normed $\mathbb R$-vector space and $(X_t)...
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33 views

If $Z = 1$, $Y$ is Rademacher, $W \sim N(0, 1)$, and $X = WY$, is $X$ conditionally independent of $Y$ given $Z$?

Suppose $X, Y, Z, W$ are random variables. If $Z = 1$, $Y$ is Rademacher, $W \sim N(0, 1)$, and $X = WY$, is $X$ conditionally independent of $Y$ given $Z$, i.e., $f(x, y|z) = f(x|z)f(y|z)$? Here, $f$ ...
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58 views

Why does the conditional independent rule of INTERSECTION require STRICT POSITIVE DISTRIBUTION?

Recently, I was confused with the proofs of some conditional independent rules (decomposition, weak union, contraction, intersection), particularly the conditional independent rule of INTERSECTION. In ...
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52 views

Question on independence of multivariate Gaussian under orthogonal projections onto subspaces

Consider $X:=(X_{1},...,X_{n})$ multivariate Gaussian distribution, i.e. $X \sim \mathcal{N}_{n}(\mu, \Sigma).$ Now let $P_{E_{1}}$ and $P_{E_{1}^{\perp}}$ denote the orthogonal projection onto $E_{1}$...
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59 views

Probability that no consecutive heads or tails occur in a sequence in which exactly 10 tails occurred and the last outcome is a tail

I am solving a problem given in Hugh Gordon's "Discrete Probability" book (Section 3.1, Problem 10) Problem A coin is tossed repeatedly until tails has occurred ten times. a. What is the ...
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33 views

Does S form a basis for R2?

I have a question here i can not explain my answer if it is right i do not sure Let S = {A = (a1, a2) , B = (a2, b2)} be a spanning set for R2 and some element x = (a, b) ∈ R2 We have x = c1 A + c2 B ...
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39 views

Random walk of $k$ particles on a $n$-dimensional hypercube

I would appreciate your help with the following, if possible. Consider an $n$-dimensional hypercube where nodes are connected by an edge if they differ in a single bit. There are $k=poly(n)$ particles,...
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87 views

Formula of the probability of two dependent events

I'm wondering if there is a formula for the probability of two dependent events $A$ and $B$. I know that if they are independent, the formula is: $$ P(A \cap B) = P(A) \, P(B) $$ maybe if they are ...
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75 views

Step of Kolomogorov 0-1 law proof

I'm trying to understand the proof of the Kolomogorov 0-1 law, and I'm stuck on the following lemma: Let $A_1, A_2, ..., B_1, B_2, ...$ be a collection of independent events. then $\sigma(A_1, A_2, .....
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62 views

Finding the probability of an $n$-th day being dry given independence and constant probability assumptions

Problem I am trying to solve the following question (3.34 from Paul Meyer's "Introductory probability and Statistics", 2nd ed.): The following (somewhat simple-minded) weather forecasting is ...
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100 views

Example of 2 random variables s.t. $(X+Y)$ ~ $U(0,2)$

In my book I found: Can you give an example of 2 random variables $X,Y$ S.T $(X+Y)$ ~ $U(0,2)$ and $X,Y$ are not independent. Any ideas of how I can find such 2 random variabes? I would prefer if ...
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82 views

If $X \perp Y|Z$, does this mean that $X \perp Y|Z, W$? How about the other way around?

Suppose $X, Y, Z, W$ are random variables. Let $\perp$ denote independence. $f$ denotes the probability density function. For example, $f(X|Z)$ is the conditional pdf of $X$ given $Z$. Does $X \perp Y|...
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61 views

Convergence of Fourier series: which kind of convergence is being used here?

I'm trying to understand the proof of Lemma 9 from Sprindzhuk's book Metric Theory of Diophantine approximation. Here's what it says. Lemma 9. Fix integers $m$ and $n$ and consider the tori $\mathbb{T}...
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44 views

Example of a discrete-time martingale with uncorrelated, but dependent, increments

Can you give me an example for a martingale with discrete time parameter, where the increments are not independent, just uncorrelated?
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35 views

Simple proof linearly dependent column imply linearly dependent rows

I am wondering if there is an easy/simple way to show that if for a given a matrix $A$ there is a non-zero vector $x$ such that $Ax=0$ then there is a non-zero vector $y$ such that $y^TA=0$ What I ...
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15 views

Closure of independence: generated sigma algebra by independent family of events

I saw this question: Independence of events and sigma algebras generated by these events, which made me think whether independence is closed under arbitrary set operations for a given family of ...
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34 views

Pdf of sum of independent rvs is the convolution of pdfs proof

I am trying to prove the statement in the title. However, I want some help to make my derivation mathematically rigorous. I have $X_{1}$ and $X_{2}$ which are two independent rvs. In addition, I have $...
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33 views

Orthogonal transformation of multivariate Bernoulli-Gaussian distribution

Recently, I studied multivariate Bernoulli-Gaussian distribution which is very useful for sparse signal processing. Suppose $X = (X_{1}, \cdots, X_{n})$ are i.i.d BG($p, \sigma^{2}$), we can know that ...
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78 views

Prove that $ \min\{X,Y\}$ and $\mathbb{1}_{\{X \leq Y\}}$ are independent [duplicate]

Suppose $X\sim\text{Exp}(\lambda_1)$ and $Y\sim \text{Exp}(\lambda_2)$ are independent. Define $Z := \min\{X,Y\}$ and $S:= \mathbb{1}_{\{X \leq Y\}}$. I want to prove that $Z$ and $S$ are independent. ...
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Equivalence between joint and conditional probabilities

I am studying conditional and joint probabilities, and a particular thought has me stumped. In a scenario with three different variables with binary outcomes, A, B, C and no independence or dependence ...
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32 views

Sequence of random variables depending on another random variable

I am working on the following problem: Suppose that $U\sim\rm Unif[0,1]$ and consider a sequence of random variables $X_i$ (which are iid when the value of $U$ is given) with $X_i \sim \rm Ber(U)$ (i....
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Is this problem well defined?

Suppose that we have the following probability distributuions $q$ is the probability that gives positive measure in any state $\omega\in\Omega$ where $\Omega$ is a fixed and finite state space. $p$ ...
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146 views

Independence of $A$ and $B$ implies the independence of $\neg A$ and $B$

Does the following apply? $$P(A\mid B)=P(A)\implies P(\neg A\mid B)=P(\neg A)$$ My rough answer is that suppose $A$ is the probability of rainy and $B$ is the probability of toothache. Then both $P(A\...
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dimension of $M = \{ x \in \mathbb{C}^{n} \ | \ \sum_{i=1}^n x_i=0 \}$

What is the dimension of $M = \{ x \in \mathbb{C}^n \ | \ \sum_{i=1}^n x_i=0 \}$? New Attempt: Let $f$ be the linear map $f: \mathbb{C}^n \to \mathbb{C}$ defined by $f(x) = \sum_{i=1}^n x_i$, then we ...
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32 views

Facts about partial independence

I am trying to build some intuition about partial independence among several variables and I wrote down these facts to check my intuition/understanding. Do you think they are True/False ? ( ideally in ...
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44 views

Are $af(x)$ and $f(x)$ linearly independent if they are restricted to different domains?

This question arise from a year long project here: https://mathoverflow.net/questions/409087/is-there-a-treatment-to-relate-the-multiple-scale-analysis-or-scale-separation-t I'm working out some ...
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54 views

$\frac{N_1}{\sqrt{N_{1}^{2} + N_2^2}} \perp \frac{N_2}{\sqrt{N_{1}^{2} + N_2^2}}$ where $N_1, N_2 \sim \mathcal{N}(0,1)$ are independent?

I have the following situation Let $N_1, N_2 \sim \mathcal{N}(0,1)$ two independent r.v. Let $X = \frac{N_1}{\sqrt{N_{1}^{2} + N_2^2}}$ and $Y = \frac{N_2}{\sqrt{N_{1}^{2} + N_2^2}}$. Now I know how ...
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26 views

Log-Ratio of IID RVs

Suppose $X,Y \in [0,\infty)$ are i.i.d. random variables with atomless CDF $F$ and PDF $f$. Their (log-)ratio is denoted $R=X/Y$ and $S=\log{R}$. The CDFs of $R$ and $S$ are $G$ and $H$, respectively, ...
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47 views

Are these two definitions of independence of random variables equivalent?

I was taught that two random variables $\xi$ and $\eta$ are independent if and only if: $$\forall a\in\mathbb{R},\forall b\in\mathbb{R}: \mathbb{P}(\xi<a,\eta<b)=\mathbb{P}(\xi<a)\mathbb{P}(\...
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34 views

What Pi-System is used to define independence for random variables?

Independent random variables [ edit] The theory of $\pi$-system plays an important role in the probabilistic notion of independence. If $X$ and $Y$ are two random variables defined on the same ...
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10 views

Iterated conditional expectation w.r.t independent r.v.

I have a sequence of independent random variables $\{X_j\}_{j=1}^k$ and some integrable function $g$. I think it holds that the iterated conditional expectation $\mathbb{E}[ \dots \mathbb{E}[ g(X_1, \...
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61 views

References for this inequality in probability?

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Let $(X_n)_{n\ge 1}$ be a sequence of i.i.d random variables of this space which takes values on $\mathbb{N}$. We then define : $\begin{...

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