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.

1,441 questions
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Could someone explain conditional independence?

My understanding right now is that an example of conditional independence would be: If two people live in the same city, the probability that person A gets home in time for dinner, and the ...
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A simple way to obtain $\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s}$.

Let $p_1<p_2 <\cdots <p_k < \cdots$ the increasing list in set $\mathbb{P}$ of all prime numbers . By sum of infinite geometric series we have $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$, ...
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Why does zero correlation not imply independence?

Although independence implies zero correlation, zero correlation does not necessarily imply independence. While I understand the concept, I can't imagine a real world situation with zero correlation ...
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Existence of independent and identically distributed random variables.

I often see the sentence "let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with a certain distribution". But given a random variable $X$ on a probability space $\Omega$, how do I know ...
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Factoring $1+x+\dots +x^n$ into a product of polynomials with positive coefficients

Can the polynomial $1+x+x^2+\dots +x^n$ be factored, for some $n\ge 1$, into a product of two non-constant polynomials with positive coefficients? Thoughts It is easy to factor it into polynomials ...
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Two tails in a row - what's the probability that the game started with a head?

We're tossing a coin until two heads or two tails in a row occur. The game ended with a tail. What's the probability that it started with a head? Let's say we denote the game as a sequence of heads ...
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Uniform distribution on a simplex via i.i.d. random variables

For which $N \in \mathbb{N}$ is there a probability distribution such that $\frac{1}{\sum_i X_i} (X_1, \cdots, X_{N+1})$ is uniformly distributed over the $N$-simplex? (Where $X_1, \cdots, X_{N+1}$ ...
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Independence of disjoint events

I'm taking a class in Probability Theory, and I was asked this question in class today: Given disjoint events $A$ and $B$ for which $$P(A)>0\\ P(B)>0$$ Can $A$ and $B$ be independent? My ...
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Independence intuition

Toss two fair dice. There are $36$ outcomes in the sample space $\Omega$, each with probability $\frac{1}{36}$. Let: $A$ be the event '$4$ on first die'. $B$ be the event 'sum of numbers is $7$'. $C$ ...
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independent, identically distributed (IID) random variables [closed]

I am having trouble understanding IID random variables. I've tried reading http://scipp.ucsc.edu/~haber/ph116C/iid.pdf, http://www.math.ntu.edu.tw/~hchen/teaching/StatInference/notes/lecture32.pdf, ...
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A criterion for independence based on Characteristic function

Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be ...
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Requirements on fields for determinants to bust dependence.

Being very much used to working on $\mathbb R^n$ $\mathbb C^n$ I just played around a bit with $$M = \left[\begin{array}{cc} 2&1\\ 1&2 \end{array}\right]$$ If the elements are in "integers ...
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Number of successes when the successes are positively correlated

In $n$ trials, each with success-probability $p$, what is the probability of at least $k$ successes, $P[n,k]$ ? The answer depends on the dependence between the trials: A. If the trials are ...
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Prove the probability measure of $X=Y$ is $0$

Let $(X,Y)$ be a random variable that takes values in $\mathbb{R}^2$. We say $X$ and $Y$ are independent if $E(f(X)g(Y))=Ef(X)Eg(Y)$. Prove that if $P(X=a)=0$ for all $a\in\mathbb{R}$, then $P(X=Y)=0$....
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Difference between Bernoulli random variables

Given are $n$ independent Bernoulli random variables with parameters $p_1,\dots,p_n$. We want to split them into two parts so as to minimize the expectation $\mathbb{E}[|X-Y|]$, where $X$ is the sum ...
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What can be said about i.i.d. $X$ and $Y$ such that $XY=(X+Y)/2$ in distribution?

Let $X$ and $Y$ be i.i.d. If $(X+Y)/2$ is equal in distribution to $XY$, then what do we know about the distributions of $X$ and $Y$? I feel like I can't say much about these distributions. I can ...
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Is a sequence of r.v. are independent if and only if their characteristic factors as a product?

It is obvious that if $\vec{X} = (X_1,X_2\cdots,X_n)^T$ are independent random variables, and their marginal characteristic function and joint characteristic function exists, then they are related by ...
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Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R$ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
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Conditional expectation involving some complications around exponential random variables

Here is my problem. Consider four independent exponential distributions $X^A_1$, $X^B_1$, $X^A_2$, $X^B_2$ where $X^A_1$ and $X^B_1$ are $\exp(\lambda_1)$ and $X^A_2$ and $X^B_2$ are $\exp(\lambda_2)$....
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Reduced row echelon form and linear independence

Let's say I have the set of vectors $S = \{v_1, v_2, ..., v_n\}$ where $v_j \in R^m$, $v_j = (a_{1j}, a_{2j}, ..., a_{mj})$. If the matrix formed by each of the vectors $A=[v_1, v_2, ..., v_n]$ looks ...
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union of two independent probabilistic event

I have following question: Suppose we have two independent events whose probability are the following: $P(A)=0.4$ and $P(B)=0.7$. We are asked to find $P(A \cap B)$ from probability theory. I know ...
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Is it possible that two independent variables become dependent conditioning on a third random variable

Is that possible that random variables $X$ and $Y$ are independent but they are no longer independent if condition on another random variable $Z$? Is there a mathematical example and an approximate ...
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Distribution of $\max(X_i)\mid\min(X_i)$ when $X_i$ are i.i.d uniform random variables

If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$\max(x_i) \mid \min(x_i) \sim \mathrm{Uniform}(\min(x_i),b)$$ I agree that the probability ...
Prove that $\mathbb P(X>Y) =\frac{b}{a + b}$ if $X, Y$ are exponentially distributed with parameters $a$ and $b$.
Let $X, Y$ be an exponentially distributed random variables with parameters $a, b$. Then $X$ has pdf: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...