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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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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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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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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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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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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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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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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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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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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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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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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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. $p$ ...
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 $P(A\... 2answers 59 views dimension of$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$, then we ... 0answers 32 views Facts about partial independence I am trying to build some intuition about partial independence among several variables and I wrote down these facts to check my intuition/understanding. Do you think they are True/False ? ( ideally in ... 0answers 44 views Are$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 ... 2answers 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 ... 0answers 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, ... 1answer 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}(\... 1answer 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 ... 0answers 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, \...
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{...