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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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How to prove this relation for Kendall's distribution function (or Kendall's measure)

Kendall Distribution Function (Nelsen, 2006, p. 163) Or Kendall Measure (Salvadori et al., 2007, p. 148) Or Kendall Function (Joe, 2014, pp. 419–422) is the cumulative distribution function (CDF) of ...
khoshmard's user avatar
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What is the formula for PMF from adding independent zero-modified negative binomial random variables?

Let $X$,$Y$,and $Z$ be independent identical zero-modified negative binomial (ZMNB) random variables, then how would the probability mass function (PMF) be calculated? $$P(X + Y + Z = k) = ?$$ For ...
vmulay's user avatar
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$\forall x:\mathbb P(M/n\leq x\mid S=n\mu +N)\to 1$ if $N=o(n)$?

Let $(X_n)_n$ be a sequence of iid random variables with values in $\mathbb N$ and expected value $\mu$, $S_n=X_1+\ldots+X_n$, $M_n=\max(X_1,\ldots,X_n)$. Furthermore suppose that $N=o(n)$ and $\...
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independence with respect to the sum of independent random variables

Assume we have $X$, $Y$, $Z$ are jointly independent random variables. Next, assume that $$ X = X_{1} + X_{2} + X_{3} $$ and $X_{1}$, $X_{2}$ and $X_{3}$ are jointly independent. Does it follow that $...
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Change of variable in conditional density function

I am reading a paper that makes the following argument: Define $\psi_i := E[x_i | \mathcal{F}_{t-i}]$ and $\epsilon_i := \frac{x_i}{\psi_i}$, where we assume that $\epsilon_i$ is independent and ...
Residual Claimant 's user avatar
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Probability that at least one event happens (dependent events)

Problem description Assume we have a bag filled with marbles of two different colors, red and blue. Our goal is to be able to pick out at least one red marble by picking out the least amount of ...
jimkokko5's user avatar
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Lower bound on the probability that the number of occurrences from a set of pairwise independent random events is 1?

I am following an online lecture series and the lecturer leaves proving the following statement as an exercise. The statement Let $A_1,\dots,A_m$ be pairwise independent random events with $\mathbb{P}[...
zkperson's user avatar
1 vote
2 answers
73 views

How to directly calculate $P(A)^{\prime}$?

Six different coloured balls are placed in a box. Kendra and Abdul each select a ball without replacement. What is the probability that Kendra does not select the green ball and Abdul does not select ...
ryangosling's user avatar
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functions of independent random variables with shared parameter dependences

Suppose $X$ and $Y$ are independent random variables. The standard theorem found in books states that $f(X)$ and $g(Y)$ are also independent regardless of the specific form of $f$ and $g$. However, ...
user6006085's user avatar
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1 answer
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Defining random variable on given sample space

Let $(\Omega, F, P)$ be a probability space with $\Omega = \{1,2,\ldots, 26\}$ and $P$ has uniform distribution the task is to define two discrete independent random variables $X,Y$ with $X$ being a ...
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I am interested in formally verifying that $w_1(z) = 1 + \frac{1}{z}$ and $w_2(z) = \frac{1}{z} e^{-z}$ are linearly independent solutions.

When solving a differential equation in the complex space, I obtained the solution in the vicinity of $z_0 = 0$: $ w(z) = A\left(1 + \frac{1}{z}\right) + B\left(\frac{1}{z} e^{-z}\right). $ Now I am ...
user avatar
1 vote
3 answers
133 views

Unions & Intersection of Probability

I've had all three answers below marked wrong, and I am not sure how to proceed. I have included my thinking. Suppose we are interested in the buying habits of shoppers at a particular grocery store ...
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Help Understanding Theorem 2.2 Proof from Rue's Gaussian Markov Random Field

I am currently reading the book Gaussian Markov Random Fields by Rue (2005) and I am having trouble understanding the proof of Theorem 2.2. The theorem and its proof are as follows: Theorem 2.2: Let $...
clementine1001's user avatar
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1 answer
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Simple ergodic convergence proof for iid

This is just a simple question from a problem sheet. Consider a sequence of independent identically distributed random variables $Y_0,Y_1,\dots$. Let $f$ be a function such that $\mathbb{E}|f(Y_0)|^2 &...
dlanshiwen's user avatar
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How to force a product of two i.i.d. random variable to be gaussian

This question is related to this other question of mine: I realized that my original question was maybe too abitious, and I would like to discuss a much more limited version of it. Consider two real ...
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$ (1 + z^2)w'' - 2w = 0 $ in complex space

Find the linearly independent solutions of the equation $(1 + z^2)w''-2w = 0$ in the vicinity of the point $0$. Attempt: To find the linearly independent solutions to the differential equation: $(1 + ...
user avatar
2 votes
1 answer
47 views

Is $\int_{\tau_0}^{\tau_1}f(W_t)\mathrm{d}t$ independent of $\mathcal{F}_{\tau_0}$ for some stopng times $\tau_0, \tau_1$ and Lebesgue integrable $f$?

Let $W=(W_t)_{t\geq 0}$ be the standard Brownian motion, let $X_t=W_t+X_0$ with initial distribution $X_0\sim\delta_x$ for $x\in\mathbb{R}$, let $\tau_1^1=\tau^1=\inf\{t>0:X_t=1\}$, write $\tau_0^1=...
Daan's user avatar
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2 answers
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Is $\int_s^t f(W_u)\mathrm{d}u$ independent to $\mathcal{F}_s$ for all (Lebesgue) measurable $f$?

I am trying to prove (or disprove) whether $$ I[s,t]=\int_s^t f(W_u)\mathrm{d}u $$ is independent of the $\sigma$-algebra $\mathcal{F}_s$ for any $t,s$ with $t\geq s$, for $W=(W_t)_{t\geq 0}$ the ...
Daan's user avatar
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0 answers
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Independence of two argmax

Problem: Suppose I have two random variables $X_1$ and $X_2$, also I have a measurable (for each $\theta$) function $f(X, \theta)$, where $\theta$ is a scalar parameter. I know that for each $\theta \...
Grigori's user avatar
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Question about the Definition of Conditional Independence

For three random variables, $X$, $Y$, and $Z$, and they all have probability densities. We say that $X$ and $Y$ are considered conditionally independent given $Z$, $(X\perp Y|Z)$ if $$ p(y|x,z)=p(y|z) ...
叶心萤's user avatar
2 votes
0 answers
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Autocorrelation and Spectrum of a Gaussian Process

I think there is a mistake in my thinking below, where is the error? If $x(t) \sim N(0, \sigma^2)$ and is temporally independent, then the autocorrelation function is $G(\tau) = \mathbb{E}[x(t)x(t+\...
Mashe Burnedead's user avatar
1 vote
3 answers
40 views

Independence of maximums of independent random iid exponential variables.

Given Random Variables $X_1,...,X_n \sim Exp(\lambda)$, I want to show that for $m \neq n$ the sets $\{X_n = \max_{i=1,...,n} X_i \}$ and $\{X_m = \max_{i=1,...,m} X_i \}$ are independent. My first ...
undergradstudent123's user avatar
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21 views

Modified Markov chain

Let $x_t, y_t$ be to independent Markov chains with non-zero transition probabilities and two states {0, 1}. $t$ is a positive integer. $x_0 = 0, y_0 = 1$ . Process $a_t$ is defined as $a_t = c^{x_t} ...
ArtBac's user avatar
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Example of random process with independent increments and almost everywhere discontinuous variance

Could we provide any example of random process with independent increments and with existing and finite, but almost everywhere discontinuous (or at least not differentiable) variance? If not, why so? ...
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1 answer
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Prove symmetry of probabilities given random variables are iid and have (not absolutely) continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous. Denote common distribution as $$F(y) :...
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Prove independence of events given random variables are iid and have (not absolutely) continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_Y(y) := F_{Y_1}(y) = F_{Y_2}(y)...
BCLC's user avatar
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1 answer
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Conditional Independence involving four events

I am trying to verify if $A \perp C | H$ and $B \perp C | H$ implies $A\cap B \perp C | H$. So far, I did the below -- but feel I'm skipping something in the step indicated (*) below. Any help would ...
KRG's user avatar
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4 votes
3 answers
78 views

Independent random variables with $X^2 + Y^2 =1$

Does there exists independent non-constant random variables with $X^2 + Y^2 =1$? I think not because intuitively if there is a relation between them it must mean they are dependent but I can't think ...
Invincible's user avatar
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0 votes
1 answer
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Independence assumption for interarrival time [closed]

I am new to Queuing systems. There is an independent assumption made for the interarrival time. Can someone please explain to me why this assumption is true, can you provide me with an example? "...
romesh prasad's user avatar
3 votes
1 answer
84 views

Independence of Sample Mean and Sample Variance

Let $X_1, \ldots, X_n$ be independent real random variables, with $n > 1$. Define the sample mean by : \begin{equation*} \overline{X} = \frac{1}{n} \sum_{i=1}^n X_i \end{equation*} Define the ...
温泽海's user avatar
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Consistency of a set of well-founded sets that is not well-founded

In the proof that $U=V \leftrightarrow Foundation$, the $\rightarrow$ direction can be easily shown by considering $grounded$ sets ($x$ is grounded iff $\forall y: x \in y \rightarrow (y$ is well-...
Niko Gruben's user avatar
3 votes
2 answers
115 views

Does Independence hold for $\sigma$-Algebras Generated by Disjoint Subsets of an independent Sequence

I want to show that for a sequence of independent random variables $(X_i)_{i \in \mathbb{N}}$ we have that for any two disjoint sets $A,B \subset \mathbb{N}$ we have that $ \sigma(X_i : i \in A)$ and ...
MathMaestro's user avatar
1 vote
0 answers
19 views

Prove that $ \cup_{i \ge 1} X_i ^{-1} (\mathcal{B})$ is a $\pi$-system for $(X_i)_{i \in \mathbb{N}}$ iid random variables

I want to show that $ \cup_{i\ge 1} X_i ^{-1} (\mathcal{B})$ is a $\pi$-system for $(X_i)_{i \in \mathbb{N}}$ iid random variables. Where $ \mathcal{B}$ is the Borel sigma algebra on $\mathbb{R}$. If ...
user007's user avatar
  • 615
2 votes
1 answer
36 views

Prove that $Z_n$ is independent of $(Y_{i,n})_i$

Suppose we have $Y_{i,n}, i \ge 1, n \ge 1 $ iid with expectation $\mu$. And given $Z_{n+1} = \sum_{i=1} ^{Z_n} Y_{i,n}$ and $Z_0 = 1$. In lecture it was stated that $Y_{i,n}$ is independent of $Z_n$....
user007's user avatar
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0 answers
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X independent of Z and Y independent of Z imply XY independent of Z?

I have a question for which I have a somewhat unclear explanation. Namely, if $X$ and $Y$ are each independent of $Z$ (i.e., $X$ is independent of $Z$ and $Y$ is independent of $Z$), then is $XY$ ...
Perelman's user avatar
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$\sigma([X{(t_1)}\in B_1,\ldots,X(t_n)\in B_n] : t_i\in\mathbb{T}, B_i\in \mathcal{B})=\sigma(X(t):t\in\mathbb{T})$?

Often when proving that stochastic processes with general index set $\mathbb{T}\subset \mathbb{R}$, possibly uncountable, are independent it is proven that any event like $$[X{(t_1)}\in B_1,\ldots,X(...
Perelman's user avatar
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2 votes
1 answer
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$X$ $Y$ are independent so are $f(X)$ and $f(Y)$ if $f$ is continuous

Assume you have a sequence of random variables $X_1,X_2,\ldots,X_n$ taking values in a (separable) metric space $\mathcal{X}$. Moreover assume $f:\mathcal{X}\rightarrow \mathbb{R}$ is continuous. Then ...
Perelman's user avatar
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Are we always allowed to say "Consider a sequence of mutually independent Random variables"

I am working on finding a counter example to some problem. For this I have set up $Z_i \sim U(-1,1)$, $\forall i \geq 1$ and $Y$ such that $P(Y=1) = P(Y=-1)$. Now for my counter example I would need $\...
matte_studenten's user avatar
2 votes
0 answers
84 views

Probability distribution of stochastic process with independent increments asymptotically approaching Gaussian

I am independently working through van Kampen's Stochastic Processes in Physics and Chemistry. It's a rewarding book to work through, but I am having trouble with question 46 on pg. 89. Below is the ...
Bulworth's user avatar
1 vote
0 answers
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Intuition for independence

Suppose that $X_1$, $X_2$, $X_3$ are independently identically distributed real random variables. $P(X_1 < X_2 \cap X_1 < X_3) = \frac{1}{6}$ by symmetry. Thus ${X_1 < X_2}$ is not ...
Adam's user avatar
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1 vote
0 answers
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How to show that $X$ and $Y$ are independent without using the uniqueness of the MGF

Let $(X, Y)$ be an absolutely continuous random vector and $M_{X, Y}$ be the bivariate moment generating function of $(X, Y)$. I want to show that $X$ and $Y$ are independent if and only if $$M_{X, Y}\...
Cyclotomic Manolo's user avatar
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0 answers
13 views

Interchanging infinite sum and limit in distribution

I'm trying to do a proof for a project and I've run into the following problem. For each $j$ consider a sequence $(Y_{j,n})_{n \in \mathbb{N}}$ of random variables such that the different sequences ...
Snildt's user avatar
  • 376
2 votes
1 answer
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Equivalent definition of independent increments of a stochastic process.

Let $(X_t)_{t\geq0}$ be a stochastic process on some probability sprace $(\Omega, \mathcal{F}, P)$. Then for $s < t$, we define the $\textit{increment}$ of the process, $X_t - X_s$ over the ...
VlakecTomaz's user avatar
3 votes
1 answer
172 views

How to get this formula for expectation of continuous-time urn process

We define the continuous-time, multi-type branching process $(X(t))_{t\ge0}$ as follows: $(X(0))=\alpha\in\mathbb{R}^d$, where $\alpha=(\alpha_1,\dots,\alpha_d)$ is the urn initial composition, ...
Dada's user avatar
  • 701
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0 answers
57 views

Conditional likelihood, conditional independence and joint independence

Consider a sequence of data samples generated from $n$ independent random vectors $(X_1, Y_1), (X_2,Y_2), (X_3,Y_3) ...$ $$D = (x_1,y_1), (x_2,y_2), (x_3,y_3) ...$$ Where $(X_i, Y_i)$ - is a random ...
spie227's user avatar
  • 21
6 votes
1 answer
525 views

A false "proof" that record setting events are dependent

Let $\{X_i\}$ be a sequence of i.i.d continuous RV. Call $i$ record-setting if $$X_i > \max_{1 \leq j < i} X_j.$$ It is well-established on math.SE and elsewhere that the events "$X_n$ is ...
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Proving pairwise independence for random matrix hash functions

Let $H_n^m$ be the family of hash functions with $h(x) = Ax + b$ for $A \in \mathbb{Z}_2^{m \times n}$ and $b \in \mathbb{Z}_2^m$. I'm trying to prove that this family of hash functions is pairwise-...
Germ's user avatar
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1 answer
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Is it possible to find 3 pairwise independent events in a space of six points?

I'm having trouble with exercise 4.6 in Billingsley, I can't seem to find 3 pairwise independent events (excluding the trivial events, the empty set and whole space) in a space of six equally likely ...
Bubble03's user avatar
0 votes
1 answer
40 views

If $X, Y$ independent, and $Y$ has same distribution as $Z$, are $X, Z$ independent?

$$ \newcommand{\Prob}{\mathbb P} $$ Let $(E, \mathcal E, \mathbb P)$ and $(F, \mathcal F)$ be a probability and measure space, respectively. Suppose we have the random variables $X, Y, Z \colon E \to ...
caitlin's user avatar
  • 125
0 votes
1 answer
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Probability that random calls are made within 5 minutes of each other

Problem from statistics textbook: Two telephone calls come into a switchboard at random times in a fixed one-hour period. Assume that the calls are made independently of one another. What is the ...
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