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.

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23 views

Coin with probability p (Biased?)

\begin{array}{l}{\text { Suppose a coin is tossed three times independently, with probability of land- }} \\ {\text { ing heads } 0 \leq p \leq 1 \text { and a complement probability } 1-p \text { of ...
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Conditional Expectation on Several Variables [on hold]

The question is from a note of econometrics. For a linear model $Y_t=X_t'\beta +\varepsilon_t$: Assumption 1: {$Y_t,X_t'$}$_{t=1}^m$ is an i.i.d random sample. Assumption 2: $E(\varepsilon_t|X_t)=0$....
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How do I prove that these are independent random variables?

Let $\{X_n\}$ $(n\geq 0)$be a sequence of independent random variables such that $0\leq X_n\leq 1$ and $\sum_n X_n$ is pointwise convergent. Then, how do I prove that $\{X_0,X_1,\sum_{n=2}^\infty X_n\...
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27 views

Stuck on a probability problem, need to find an easier solution

I'm currently studying probabilities and I'm stuck on the following problem: We collect cards of 3 different types, each time that we get a card, it is of the type "i" with the following ...
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1answer
15 views

Example of independence of functions of dependent random variables

Is it true that it is not possible for functions of dependent random variables to be independent? For example, if $X_1, ..., X_n$ are dependent, then it is impossible for $Y_1 = X_1^2, ..., Y_n = X_n^...
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30 views

Probability - Defective product

A company sends 30% of its product to Client A and 70% to Client B. Client A reports that 5% of the products it received are defective, whereas Client B reports that 4% of products received are ...
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1answer
45 views

Are $A|B$ and $B$ independent events? [on hold]

Suppose $A$ and $B$ are two dependent events, that is $P(A\cap B)>0$. We know that $P(A\cap B)=P(A|B)P(B)$. Is it true that $A|B$ and $B$ are independent? From my understanding, two events $X$ ...
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1answer
43 views

Show that if $X_1, \dots, X_n$ are i.i.d., then two expectations are equal.

Let $X_1, \dots, X_n$ i.i.d random variables. Put $S_n:= \sum_{k=1}^n X_k$. Show that $\mathbb{E}[X_1 I_{\{S_n \in A\}}]= \mathbb{E}[X_j I_{\{S_n \in A\}}]$ for $1 \leq j \leq n$, where $A$ is an ...
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1answer
24 views

If the probability of current flowing in circuit is known, how can I know the probability that a certain bulb will work?

Here the schema is very important: The probability that a bulb will work is 0,5. The probability that the current will flow in circuit is 0,3984375. What is the probability that the bulb C will work....
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1answer
26 views

What is the probability that the current will flow in the circuit, if P(1/2, that the bulb will work) and there are 7 bulbs.

So, the problem is as follows: Calculate the probability that the current will flow in circuit if the chance that a light bulb will work is 0,5 and there are totally 7 bulbs. Here is the schema: I ...
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1answer
32 views

$P(\limsup A_n)=1 \Leftrightarrow \sum_{n=1}^{\infty} P(A \cap A_n) = \infty\; \forall A, P(A)>0$

Let $\{A_n\}$ be a sequence of independent events. How to prove that $$P(\limsup A_n)=1 \Leftrightarrow \sum_{n=1}^{\infty} P(A \cap A_n) = \infty\; \forall A, P(A)>0?$$ As the $A_n$ are ...
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1answer
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Solution of linearly dependent functions

I'm having a lot of trouble with this question. I know they are not linearly independent, but I'm not sure how to proceed. Here is the problem: Thank you.
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Independence of a Random Variable from a Sigma Field and Expectation

The question is: "Show that $X$ is independent of $\sigma(Y)$ if and only if for bounded and measurable $f,g$, $E[f(X)g(Y)]=E[f(X)]E[g(Y)]$." I think I have managed to prove the forward statement by ...
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1answer
25 views

Independence of a convergent series of i.i.d. random variables

Suppose $\{\epsilon_n \vert n \in \mathbb{Z}\}$ is an i.i.d. collection of $L_2$ random variables and $\sum_{i=0}^\infty \vert \varphi_i\vert < \infty $ such that the time series $ \sum_{i=0}^\...
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1answer
15 views

Measure Theory - Expectation and Independence

Can we use $E[f(X)g(Y)]=E[f(X)]E[g(Y)]$ for $f,g$ bounded and measurable to show that $X$ is independent of $\sigma(Y)$?
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Independence of random variables given an event

Suppose we have independence of two random variables, $X$ and $Y$. This of course then implies that $p(x,y)=p(x)p(y)$ where $p(x,y)$ is the joint pmf of X and Y (ie. $p(x,y) = P(X=x, Y=y) = P(X=x)P(Y=...
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3answers
38 views

Linear dependence of functions involving $e^x$

On a test today I was given the following functions: $$ f(x) = (-1+x)e^x$$ $$g(x)=-2e^x$$ We were asked to show if it was linearly dependent or linearly independent So I showed that if I ...
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1answer
24 views

Calculating the expected value of the amount of same numbers chosen by two people.

so I've been stuck all day on one question and I have no idea what to. This is the problem: Two people choose from a set of integers ranging from 2 to 100.(so 99 different integers) One person ...
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1answer
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Independent Events and Bernstein Paradox for n events [closed]

Is it possible to extend Bernstein Paradox example (about pairwise independence, but joint dependence of 3 events (color sides of tetrahedron)) to n events using the same reasoning?
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1answer
57 views

Are $A,B,C$ independent given that $P((A \cap B )\cup C)=P(A)\cdot P(B)\cdot P(C)$

Can someone help me shading light on this question about independence?The answers look conflicting. Are these 3 events independent? Not Solved yet. Can anyone help?
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Does converse implication for PGF of sum of independent random variables hold?

Let $X_1, X_2$ be i.i.d. random variables. Then the probability generating function $G_{X_1+X_2} = G_{X_1} G_{X_2}$. Does the converse implication hold? It is quite easy to check that it does for ...
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1answer
30 views

Independence of Brownian Motion and $\mathscr{F}_{0}$

Assume $ B=\left\{ B_{t},\mathscr{F}_{t}:0\le t<\infty\right\} $ is a standard 1-dimensional Brownian motion. Then show that $\mathscr{F}_{\infty}^{X}$ and $\mathscr{F}_{0}$ are independent (...
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1answer
19 views

Independence with single random variables implies the independence of collection.

On probability space $\left(\Omega,\mathscr{F},P\right)$, assume a sub $\sigma$-algebra $\mathscr{F}_{0}$ and random variable $X,Y$ are independent with $\mathscr{F}_{0}$. Do we have $\left(X,Y\right)$...
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Example: the probability of the intersection of three events factorizes but pairwise independence does not hold

Let us consider three events A,B,C. We know that they are mutually independent iff: 1) $$P(A\cap B) = P(A)P(B)$$ $$P(B\cap C) = P(B)P(C)$$ $$P(C\cap A) = P(C)P(A)$$ 2) $$P(A\cap B\cap C) = P(A)P(B)...
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1answer
26 views

If $X$ and $Y$ are independent, then $X$ and $W$ are independent

Let $X$ and $Y$ be independent random variables. Suppose $W$ is a random variable such that $W=Y$ almost surely. Are $X$ and $W$ independent? Initial Work: Define $A=\{ \omega \; | \; Y(\omega) \ne W(...
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71 views

Are those events independent?

Someone edited wrongly my question, this is the correct version: Three events A, B, C satisfy the following condition: $$P(A \cap (B \cup C)) = P(A)P(B)P(C)$$ Are they independent? To me ...
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3answers
116 views

Are these 3 events independent? Not Solved yet. Can anyone help?

Let us consider 3 events A,B,C such that: $$P((A \cap B )\cup C)=P(A)*P(B)*P(C)$$ Notice that the second term is a union and not an intersection Are they independent? And what if the assumption ...
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Is there terminology around a bivariate function that is dependent on only one of its two arguments?

For example, if $f$ can be defined by $f(x, y) = g(x)$ for some $g$ and all $x$, are there any special adjectives that are applicable to $f$?
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IID assumption for time series data

How common or reasonable is to assume time-series data are independent and identically distributed? To do some theoretical analysis, I need the assumption but I am not sure about its validity.
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2answers
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Unfair Coin Toss probability, Independence

We have two coins. The first one is a fair coin and has $50\%$ chance of landing on head and $50\%$ of landing on tails. for the other one it is $60\%$ for head and $40\%$ for tails. We choose one of ...
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0answers
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Poisson Distribution with Dependent Events

Problem: The average number of cars arriving at a tollbooth per minute is $\lambda$ and the probability of k cars in the interval $(0,T)$ minutes is: $$P(k;0,T) = e^{-\lambda T}\frac{[\lambda T]^k}{...
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1answer
31 views

Why is $B_{t}-tB_{1}$ independent of $B_{1}$

Let $(B_{t})_{t \in [0,1]}$ be a brownian motion. Show that $B_{t}-tB_{1}$ is independent of $B_{1}$ My idea: I need to find a linear transformation $A$ that renders $A \begin{pmatrix} B_{1}-B_{t} \\ ...
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1answer
33 views

Prove random variables are pairwise independent

From Durrett's book I've come along with this problem: Let $K\geq 3$ be a prime and let $X,Y$ be independent random variables that are uniformly distributed on $\{0,1,\ldots , K-1\}$. For $0\leq n &...
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0answers
45 views

Conditional Independence for XOR

Let's say we have a basic XOR table with input variables X and Y and output Z. Could I assume that X and Y are conditionally independent? I think so, because by knowing Z, if I knew X I would also ...
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1answer
41 views

Interchange between expected value and infinite summation (Fubini theorem)

Let $S_n = \sum_{i=1}^nX_i$ (where the $X_i$ are i.i.d.) and let N be a positive, integer valued r.v., independent from the sequence $X_n$. Suppose also that $E[N]<\infty$ and $E[|X_i|]<\infty$. ...
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Sum of $k$-wise independent $p$-stable random variables

For $p\in(0,2]$, let $\mathcal{D}$ be a strict $p$-stable distribution (link). For $p=1,2$ would correspond to Couchy and Gaussian distributions respectively. By definition, if $Y,X_1,\dots,X_n\sim\...
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1answer
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Why is independence stronger than zero covariance?

For two variables to have zero covariance, there must be no linear dependence between them. Independence is a stronger requirement than zero covariance, because independence also excludes nonlinear ...
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1answer
44 views

Are two orthogonal linear transformations of the same random Gaussian vector independent?

Major revision: I want to prove/disprove the following claim: If $e = (e_{1},\dots, e_{n})$ is a vector of independent random variables, (each $e_{i}$ is normally distributed), and $v_{1}, \dots ...
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1answer
19 views

Independence between events involving three random variables

Let $X,Y,Z$ be three independent random variables, we want to find out if the following holds: $$P(X\geq Y,X\geq Z) = P(X\geq Y)P(X\geq Z)$$ that is, if $X\geq Y$ and $X\geq Z$ are independent events. ...
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1answer
21 views

Independence via conditional expectation and characteristic function

I have encountered the following statement but can't think of a proper argument to explain/prove it: If for every $A \in \mathcal F$ s.t. $P(A>0)$ we have $ E\left[ \exp (i \lambda X) | A \right] =...
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1answer
34 views

Prove or disprove two simple statements on conditional probability

STATEMENT 1: if $P[A|B]=P[B]$ then $A$ and $B$ are independent. My attempt: Assuming that $A$ and $B$ are independent then, since $P[A|B]= P[B]$ is possible only if $P[A]=P[B]$, the statement is ...
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1answer
18 views

What are some theorems giving lower bounds on the size of a maximum indepedent set of verticies of a planar graph?

Suppose that $G$ is a planar graph. What is an example of a theorem that will give us a lower bound on the size of a maximum independent set of vertices in $G$? The following may or may not be true, ...
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1answer
24 views

variance of quadratic forms when the random variables are not normally distributed

Let $A$ be a symmetric matrix and $X = (X_1,...,X_n)^T$ a vector of independent identically distributed random variables. The random variables are assumed not to be normally distributed. What ist the ...
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1answer
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Prove that strange coloring is related to maximum independent sets

Consider an un-directed simple graph $G$. Define a “tinge” on graph $G$ to be a mapping from $V(G)$ to $\{$“red”, “black”$\}$ Given a graph $G$ and a tinge $T$, we say that the black graph of $G$ ...
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3answers
49 views

Discrete Independent random variables X and Y

Let $X$ and $Y$ be independent random variables, taking values in the positive integers and having the same mass function $f(x)=2^{-x}$ for $x=1,2,..... \infty$ Find $P(Y\gt X)$. Here's what I did,...
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1answer
24 views

Multiple Random Variables Union and Intersection Complement Operation

I'm currently studying probability theory with the textbook Probability and Random Processes for Electrical and Computer Engineers (John Gubner) and had a question regarding multiple random variables. ...
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1answer
24 views

Prove partial uncorrelation implies independence, and vice versa in the special case of Gaussian random variables

Show that if $X, Y $and $Z$ are (univariate) Gaussian random variables then $\rho_{X,Y |Z} = 0$ if and only if $X\perp Y | Z$ (X is independent of Y given Z). How can we prove this using the fact ...
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1answer
39 views

Symbolic representation of orthogonality/independence in proofs

I have a very specific question concerning notation I have seen used in proofs in econometric journals. I want to home in on the difference(s) between $\perp$ versus ${\perp\!\!\!\perp}$. On the ...
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2answers
36 views

Dice rolling and independence

Two dice are rolled. $X$ is the smallest and $Y$ is the largest value that can be obtained on rolling the dice. Are $X$ and $Y$ independent? How to check that 2 joint probability mass function are ...
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1answer
43 views

Formal proof of fair coin simulation from biased coin

Von Neumann's trick to simulate a fair coin from a biased coin is well-known: Toss the biased coin twice; If you get Head-Tail, return 1; If you get Tail-Head, return 0; Otherwise, go to 1. It is ...