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 (probability) or (probability-theory). Do not use for linear independence of vectors and such.

0
votes
0answers
22 views

Three random variables independence

Is it possible to have such random variables $X, Y, Z$ (for simplicity let them be discrete) that: $$P(X \cap Y) = P(X)P(Y)$$ $$P(X \cap Z) = P(X)P(Z)$$ $$P(Y \cap Z) = P(Y)P(Z)$$ but $$P(X \cap Y \...
0
votes
1answer
16 views

Simple Exploitation of Markov Property for Brownian Motion

The elementary Markov Property as I learned it reads: For a positive real number $a$ the stochastic processes $(B_t)_{0 \leq t \leq a}$ and $(W_t)_{t \geq 0}:=(B_{t+a}-B_a)_{t \geq 0}$ are ...
0
votes
1answer
19 views

how to explain $X$ and $Z$ Not relevant, how to prove it?

Let the random variables $X$ and $Y$ be independent,$X$ follows the exponential distribution with parameter $1$, and the probability distribution of $Y$ is $$\mathcal{P}\{Y=-1\}=p,\;\mathcal{P}\{Y=1\}...
5
votes
0answers
54 views

Can sum of pairwise independent random variables be constant?

If we consider a set of random variables $X_1$ up to $X_n$ that are independent, then their sum cannot be a constant (given that at least one of them is non-constant). That is not too hard to prove. ...
-1
votes
1answer
26 views

Probability generating function with and without condition [closed]

Let $L_i, i=1,2$ conditionally independent and $L=L_1+L_2.$ $\mathbb P(L_i=k|P=p)=p(1-p)^k, k=0,1,2,..., p\in (0,1).$ $P = \begin{cases} \frac{1}{3}, & \text{with probability} \frac{1}{2},\\[2ex]...
1
vote
1answer
16 views

Conditional independence equivalent definition

Consider a probability space $(\Omega, \mathbb{F}, P)$. Let $\mathbb{F}_1$, $\mathbb{F}_2$ and $\mathbb{F}_3$ be sub-$\sigma$-algebras of $\mathbb{F}$. Assume that $\mathbb{F}_1$ and $\mathbb{F}_2$ ...
0
votes
0answers
7 views

Variance of two random variables that are defined by independent variables

Let $$Λ_1 = \frac{1}{4} Λ^{(2)} + \frac{3}{4} Λ^{(4)},$$ $Λ^{(2)}, Λ^{(4)}$ are independent random variables. $\mathbb EΛ^{(s)}=\frac{s+5}{s^2}$ and $\mathbb DΛ^{(s)}=\frac{(s+5)^2}{s^2}.$ From the ...
1
vote
1answer
12 views

Condtional Covarianz Problem

i have on Friday a presentation and i have one small problem. Let $ X_1,...,X_n$ be independent Random Variables. Let $f: \Omega^n \rightarrow \mathbb{R}$ and define $Z:=f(X_1,...,X_n)$. Define also: \...
0
votes
0answers
16 views

Bernoulli vs. binomial: Which one to apply in practice?

After reading this and this thread, I got a little confused about the distinction between the Bernoulli and the binomial distribution in practice. From what I read, I understood that: A trial (e.g. ...
0
votes
1answer
31 views

Infinite sums of independent random variables

Question: Suppose we have an independent family of random variables $$ \{X_{ij} \mid i \in \{1,\dots,m\}, j \in \mathbb{N} \} $$ on the same probability space $(\Omega, \mathcal{A}, \mathbb{P})$. ...
-3
votes
1answer
47 views

Linear algebra questions: True or False [closed]

5 vectors in $\mathbb{R}^6$ are always dependent? If $A$ is singular $n \times n$ matrix, $A^T A$ is also singular? If $P$ is a permutation matrix, then $P$ must be singular? A remedy for the accurate ...
0
votes
1answer
20 views

Prove expectation of independent R.V.s. are independent

Let $X_1, X_2, \dots$ be independent random variables, and show that $Y_n = X_n - \mathbb{E}[X_n]$ are independent
4
votes
1answer
47 views

Need help fixing/clarifying my thinking about iid RVs after learning some 1st Yr Stats

First a warning: this is not the most interesting question but I want to update my understanding of independence now that I'm taking 1st year statistics I often heard in my 1st year probability class ...
1
vote
0answers
32 views

Conditional i.i.d.ness of random variables

Let $X$ be a random variable with support $\mathcal{X}$. Let $r:\mathcal{X}\to\{0,1\}$. Assume the random variables in the sequence $(\epsilon_i)_{i=1}^N$ are i.i.d. conditional on $X$. Does this ...
0
votes
1answer
18 views

If $X$ is inpendent of $\mathcal{F}$, then why is for $A \in \mathcal{F}$, $\chi_{A}$ and $X$ independent

As the title suggests, let $X$ be a real-valued random variable and $\mathcal{F}$ a given $\sigma-$algebra. Let $A \in \mathcal{F}$, then it immediately follows that $\chi_{A}$ and $X$ are ...
1
vote
1answer
26 views

Property of pairwise independent but not independent events

Let $(\Omega, \mathcal A, P)$ be a probability space and $A,B,C \in \mathcal A$ with the properties a) $A,B,C$ are pairwise independent, b) $A\cap B \cap C = \emptyset$, c) $P(A) = P(B) = P(C) =:...
0
votes
1answer
26 views

Show the independence of three vectors

I took the following question from Zhang, Fuzhen. Linear Algebra (Johns Hopkins Studies in the Mathematical Sciences) . Johns Hopkins University Press. Show that $\alpha_1 = (1,1,0)$, $\alpha_2 = (...
-2
votes
1answer
54 views

How to understand and to prove that any $n$ linearly independent vectors form a basis of $R^n$ space conceptually?

I know how to use methods like system of linear equations to show that any vector in $R^n$ can be expressed in the form of unique linear combinations of a given set of n linearly independent vectors, ...
1
vote
1answer
33 views

Two vectors are linearly independent?

Let $x, y, z$ be vectors in vector space $V$. Suppose $z \notin L(x,y)$ , where $L(x,y)$ is the linear span of $x, y$. Show that $x, y$ are linearly independent iff x+z, y+z are linearly independent. ...
0
votes
2answers
41 views

Example of an experiment in which A, B, C are independent, but not pairwise independent

Can somebody give an example of process in which we have at least three events A, B, C and: P(A ∩ B ∩ C) = P(A) * P(B) * P(C) But A, B, C are not pairwise independent
1
vote
1answer
28 views

Events are independent or not independent in the following example?

Question. Let $X_1,X_2,X_3$ denote the outcomes of three rolls of a six-sided die. (I.e., each $X_i$ is uniformly distributed among ${1,2,3,4,5,6}$ and by assumption they are independent.) Let $...
0
votes
0answers
12 views

Definition of asymptotically independent random variables reference

I am wondering if there exist really a concept of asymptotic independance of random variables. If so, could you give me a reference (book with the number of the page where I can find this definition) ...
0
votes
1answer
29 views

Independence of random variables: sufficient conditions

I would like your help to understand which assumptions are sufficient to get a desired conclusion about independence between random variables. General setting: Let $\mathcal{T}\equiv \{1,...,T\}$. ...
1
vote
1answer
24 views

What is independence of events (in case of tossing an unbiased coin 3 times)?

An unbiased coin is tossed 3 times in a row. Define the events $A, B, C$ such that $A = \{HHH, TTT\}, B =\{TTT, TTH, THT, HTT\}$ and $C = \{HHH, TTT, HHT\}$ Now clearly $P(A\cap B) = P(A)P(B)$ and $P(...
0
votes
1answer
27 views

Switching from $\sum\limits_{y \in E}\sum\limits_{n \geq 0}$ to $\sum\limits_{n \geq 0}\sum\limits_{y \in E}$

Let $(X_{n})_{n \in \mathbb N}$ be a Markov Chain How can I show $\sum\limits_{y \in E}\sum\limits_{n \geq 0}P^{x}(\tau_{x}>n,X_{n}=y)\Pi(y,z)=\sum\limits_{n \geq 0}\sum\limits_{y \in E}P^{x}(\...
1
vote
1answer
22 views

Bounds for the number of independent variables on a finite probability space

The following question looks somehow academic but I didn't find much information on the Internet. Consider a finite probability space $\Omega=\{\omega_1, \dots, \omega_n\}$ with a probability $\...
0
votes
1answer
29 views

Establishing independence between seemingly complementary random sums

I am struggling to work through the following problem. Let $\{X_{n}\}_{n\geq 0}$ be i.i.d. random variables, $M$ a Poisson random variable with parameter $\lambda$ that is independent of the random ...
0
votes
0answers
16 views

Prove that $X_t = 1/c B_{c^2 t}$ with $c \gt 0$ is a Brownian Motion

Let $\{ B_t \}$ a standard Brownian Motion, prove that the next process are Brownian Motions: a) $\{ X_t \}$ where $X_t = -B_t$ b) $\{ X_t \}$ where $X_t = 1/c B_{c^2 t}$ with $c \gt 0$ c) $\{ ...
0
votes
1answer
34 views

Will these two random variables be independent?

Assume you have two continuous random varaibles $X,Y$. Also assume that their joint probability density function can be written $f(x,y)=p_1(x)p_2(y)$, must they then be independent? The problem is ...
0
votes
1answer
24 views

Moment generating function of two variables

I am able to do all the parts except the very last. I have been trying to coax the differential equation $\frac{M'}{M}=t$ or something to that effect but I don't see how I can achieve this. Hints ...
1
vote
1answer
42 views

How can I prove that these events are independent?

I have a pair of sets: $A=\{n\in\mathbb{N}\mid p\cdot n\}$ $B=\{n\in\mathbb{N}\mid q\cdot n\}$ Where $p$ and $q$ are two different prime numbers. And the following event definitions: $X_n$: $n\in{...
1
vote
0answers
18 views

Help me elucidate what “the vectors $(B(d):d \in \mathcal {D_n })$ and $(Z_t:t \in \mathcal {D-D_n})$ are independent” mean.

The following is taken from a proof of Lévy's construction of Brownian motion in a book by Peter Mörters and Yuval Peres. $\mathcal {D_n } := \{\frac {k } {2^n } :1 \le k \le 2^n \} $, the set of ...
1
vote
0answers
27 views

the independence property of conditional expectation

I read a proof of following property: suppose $X$ is a random variable on the probability space $(\Omega,F,P)$, $A$ and $B$ are sub $\sigma$-field of $F$, $B$ is independent of $\sigma(X,A)$, then $E(...
1
vote
1answer
17 views

For a sequence of experiments where each $X$ is the number of trials until success with varying $p$, is each $X$ independent?

Assume that, every time you buy a box of Wheaties, you receive a picture of one of the $n$ baseball player. Let $X_k$ be the number of additional boxes you have to buy, after you have obtained $k-1$ ...
0
votes
1answer
31 views

Applying the definition of mutual independence to outcomes of random variables

My book defines mutual independence as: events ${A_1, A_2, ..A_n}$ are mutually independent if for any subset ${A_1, A_2, ..A_m}$ (where $m \leq n$) of these events we have: $$P(A_1 \cap A_2 \cap ......
2
votes
1answer
59 views

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables.

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables. My attempt: Fix $A \in \mathcal{R}$ (a Borel subset of the real ...
0
votes
1answer
25 views

Conditional Independence and product of random variables

I am stuck at the following situation: Let random variables $Y, X, W_1, W_2$. I know that $W_1$ and $W_2$ are each independent from $Y$ conditional on $X$: $$p\left(Y\mid \{X,W_1\}\right) = p\left(...
1
vote
1answer
15 views

Does time changed brownian motion have no-memory property?

Let $W=(W_t)_{t \geq 0}$ be a Browniwn motion. Do the processes $$X_t = W_{e^t} \quad \text{and} \quad Y_t = \exp \left(- \frac{t^2}{2} \right) W_{e^t}$$ have the no-memory property, i.e. are the sets ...
1
vote
0answers
26 views

Two independent geometric variables [duplicate]

Suppose that $X$ and $Y$ are independent, identically distributed, geometric random variables with parameter $p.$ Prove that $P( X = i | X + Y = n) = 1/(n-1)$ for $i = 1,2,\dots ,n-1.$ So I said $$P(...
0
votes
0answers
13 views

Conditional expectations and conditional independence

While learning causal models, I came across the below assertions: to express the conditional independence of Y and X given Z: $$ Pr(Y|X, Z) = Pr(Y|Z) $$ we can write: $$ E(Y|X, Z) = E(Y|Z) $$ If X, ...
0
votes
1answer
38 views

Expected value in a linear combination

I have a random variable Y, that is defined by: $$Y = aX_1 + bX_2$$ Where we know $X_1 $ and $X_2$ are independent. How do I write out $EX_1$ and $EX_2$ in terms of only a, b, EY, and VarY? I ...
0
votes
4answers
32 views

Proof linear dependency for vectors in $3$-D space

How to proof the linear dependency / independency ONLY using vectors (not through matrixes), as I am not familiar with this concept for now. The example is the following: Are the following vectors ...
0
votes
1answer
22 views

About the definition of independent random vectors

I just have a question concerning the definition of independent random vectors. The random vectors $X=[X_1, \ldots, X_n]$ and $Y=[Y_1,\ldots, Y_m]$ are independent means that $X_1,...,X_n,Y_1,...,Y_m$ ...
1
vote
0answers
45 views

Independence of Events and Conditional Probability

A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with ...
0
votes
0answers
7 views

Does conditional independence of random variables transfer to functions of the random variables?

Suppose $$Y,Z \perp W |X$$ That is to say $Y$ and $Z$ are conditionally independent of $W$ given $X$. For two functions $f$ and $g$, does the following it in general hold? $$ f(Y,Z,X) \perp g(W,X) ...
1
vote
0answers
12 views

Derivation of the formula for the probability of a class, given conditionally independent attributes.

The following is a formula that finds the posterior probability of a class (i.e. yes or no) given four conditionally independent attributes: $$P(c|X) = P(x_1|c)\cdot P(x_2|c)\cdot P(x_3|c)\cdot P(x_4|...
1
vote
1answer
36 views

Why are the random variables independent?

This question is about independence in the ANOVA analysis. Assume that you have $I\cdot J$ independent random variables $X_{i,j}$. We also assume that each is normally distributed with common mean $\...
1
vote
1answer
24 views

Patterns in probability trees

I'm just floating this and I'm not sure how to describe my problem precisely. So I'll lead with an example: Assume a team of 3 are chosen from 4 boys and 5 girls. Let the rv $X$ be the number of girls ...
1
vote
1answer
37 views

Independent increment vs independent sigma-algebras

Suppose we want to define a Lévy process $\{ X_t \vert \ t \geq 0\} $. Is it equivalent to demand independent increments i.e. $$ \forall n \geq 1, \forall t_n \geq t_{n-1} \geq ...\geq t_1 \geq 0: X_{...
0
votes
2answers
41 views

Probability of the intersection of two events if they are dependent

A system was to select a string uniformly at random from the set {RRRR, RS, STT} and then select a letter uniformly at random from the selected string. The system is run twice (outputting two letters)...