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

78
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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 ...
27
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2answers
1k views

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 $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$, ...
24
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5answers
23k views

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 ...
23
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3answers
2k views

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 ...
16
votes
2answers
449 views

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 ...
15
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5answers
2k views

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 ...
13
votes
1answer
3k views

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 $X_1, \cdots, X_{N+1}$ ...
13
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2answers
5k views

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 ...
12
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3answers
1k views

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 $7$'. $C$ ...
12
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3answers
31k views

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, ...
12
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2answers
5k views

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 ...
11
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3answers
828 views

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 ...
11
votes
4answers
2k views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
10
votes
3answers
871 views

Examples of pairewise independent but not independent continuous random variables

By considering the set $\{1,2,3,4\}$, one can easily come up with an example (attributed to S. Bernstein) of pairwise independent but not independent random variables. Counld anybody give an example ...
10
votes
1answer
2k views

Relation between Borel–Cantelli lemmas and Kolmogorov's zero-one law

I was wondering what is the relation between the first and second Borel–Cantelli lemmas and Kolmogorov's zero-one law? The former is about limsup of a sequence of events, while the latter is about ...
10
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1answer
358 views

The theory in probability

Consider a real-life experiment (perhaps written as a problem in a textbook): A coin is continually tossed until two consecutive heads are observed. Assume that the results of the tosses are mutually ...
10
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0answers
746 views

Distribution of the sum of absolutes values of T-distributed random variables

Where X is a r.v. following a symmetric T distribution with 0 mean and tail parameter $\alpha$. I am looking for the distribution of the n-summed independent variables $ \sum_{1 \leq i \leq n}|x_i|$....
9
votes
4answers
1k views

Can independence go one way? I.e., so that P(A|B) = P(A), but P(B|A) ≠ P(B)

As I understand it, independence of A and B can be informally established by asking whether learning something about one of those events tells you something new about the other. This must be borne out ...
9
votes
3answers
4k views

Probability sum of 5 before sum of 7

Pair of fair die are rolled (independently I hope) infinitely. Find probability sum of 5 appears before sum of 7. 2 approaches: $$P(\text{sum of 5 appears before sum of 7})$$ $$= P(\text{roll 1 is ...
9
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4answers
1k views

Showing that $n$ exponential functions are linearly independent.

I have $n$ lambdas, which are all different real and positive numbers, where: $\lambda_1 < \lambda_2 < \cdots < \lambda_n$. I then have to show that these functions are linearly independent:...
9
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0answers
515 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
8
votes
4answers
2k views

A die is rolled and a coin is flipped. What is wrong with the following reasoning?

A die is rolled and a coin is flipped. What is wrong with the following reasoning?: Let $A=\{1,3,6\}$ (event from die roll), let $B=\{H\}$ (event of heads) then, $A\cap B=\emptyset$ (since they have ...
8
votes
3answers
152 views

How can I show that $X$ and $Y$ are independent and find the distribution of $Y$?

$X_1,X_2,\dots,X_n$ is an i.i.d. sequence of standard Gaussian random variables. \begin{align}X&=\frac{1}{n}(X_1+X_2+\dots+X_n) \\[0.2cm] Y&=(X_1-X)^2+(X_2-X)^2+\dots+(X_n-X)^2\end{align} ...
8
votes
5answers
234 views

What are some interesting un-intuitive problems in probability aside from Monty Hall?

Does anyone know of some interesting and almost strange problems in probability? I know that probability is sometimes notorious for being mind-bending and un-intuitive! (Monty Hall is already an ...
8
votes
1answer
521 views

Probability of getting a seat in the train car

A train has got five train cars, each one with N seats. There are 150 passengers who randomly choose one of the cars. What is the probability that everyone will get a seat? I think that what is ...
8
votes
0answers
695 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ $$\...
7
votes
2answers
746 views

How can we apply the Borel-Cantelli lemma here?

Let $(A_n)$ be a sequence of independent events with $\mathbb P(A_n)<1$ and $\mathbb P(\cup_{n=1}^\infty A_n)=1$. Show that $\mathbb P(\limsup A_n)=1$. It looks like the problem is practically ...
7
votes
2answers
83 views

Is it true that $\lim_{n \to \infty} {(P(\forall i,j\leq n \text{ } [X_i, X_j] = e))}^{\frac{1}{n}} = P(X_1 \in Z(G))$?

Suppose $G$ is a group. $\{X_n\}_{n = 1}^{\infty}$ is a sequence of i.i.d. random elements of $G$ satisfying the condition that $$\forall H \leq G, \qquad P(X_1 \in H) = \begin{cases} \frac{1}...
7
votes
1answer
3k views

Independence and conditional expectation

So, it's pretty clear that for independent $X,Y\in L_1(P)$ (with $E(X|Y)=E(X|\sigma(Y))$), we have $E(X|Y)=E(X)$. It is also quite easy to construct an example (for instance, $X=Y=1$) which shows that ...
7
votes
1answer
146 views

Show that $E(|X-Y|^{3/2})\leq 2 E(|X|^{3/2})$ when $X$ and $Y$ are i.i.d.

Let $X$ and $Y$ be i.i.d. random variables such that $E(|X|^{3/2})$ is finite. Prove that $$E(|X-Y|^{3/2})\leq 2 E(|X|^{3/2})$$ This is from a past qualifying exam. I'm really stumped by the question....
6
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4answers
5k views

What is the meaning of “independent events ” and how can we logically conclude independence of two events in probability?

What is the meaning of "independent events " in probability For eg: Two events (say A and B)are independent , what I understand is the occurrence of A is not affected by occurrence of B .But I am ...
6
votes
4answers
649 views

Variance over IID a random number of times

Let $X_1, X_2,\dotsc$ be independent and identically distributed with mean $E[X]$ and variance $VAR[X]$. Let $N$ be a non-negative integer-valued random variable independent of the $X_i$'s. Show $$ ...
6
votes
3answers
6k views

Example of Pairwise Independent but not Jointly Independent Random Variables?

I am asked to: Find a joint probability distribution $P(X_1,\dots, X_n)$ such that $X_i , \, X_j$ are independent for all $i \neq j$, but $(X_1, \dots , X_n)$ are not jointly independent. I ...
6
votes
1answer
3k views

Almost Sure convergence of sum of independent random variables

Let $\{{X_{j}}\}_{1}^{\infty}$ be independent r.v.s such that $\sum E( |X_{j}|) <\infty$. How to show that $\sum X_{j}$ converges almost surely. Can I argue simply that for every $\epsilon>0, \...
6
votes
2answers
301 views

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 ...
6
votes
2answers
53 views

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 $P(X=Y)=0$....
6
votes
1answer
171 views

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 ...
6
votes
2answers
87 views

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 ...
6
votes
1answer
39 views

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 ...
6
votes
2answers
2k views

Product of independent random variables

The following is a classic example that pairwise independent does not necessarily imply mutually independent: Let $X_1$ and $X_2$ be independent r.v.'s with distributions $$P(X_i=1)=P(X_i=-1)=\frac{...
6
votes
1answer
163 views

Almost sure convergence through subsequences

$\{X_i\}$'s are independent Poisson random variables with parameters $\lambda_i$ respectively satisfying $\sum_{n=1}^{\infty}\lambda_n=\infty$. Define $S_n=X_1+X_2+\cdots +X_n$ then show that $$\frac{...
6
votes
0answers
141 views

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 $X$ ...
6
votes
0answers
147 views

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 $\exp(\lambda_2)$....
5
votes
3answers
18k views

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 ...
5
votes
4answers
94k views

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 ...
5
votes
3answers
2k views

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 ...
5
votes
2answers
7k views

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 ...
5
votes
2answers
605 views

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: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
5
votes
1answer
671 views

Independence of complementary events

Suppose $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space, $I$ is an arbitrary index set and $\{A_i\}_{i \in I} \in \mathcal{F}^{I}$. For $i \in I$ we define $B_i^{(0)} := A_i$ and $B_i^{(1)} :=...
5
votes
3answers
1k views

Let $U=\operatorname{min}\{X,Y\}$ and $V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$.

Let $X$ and $Y$ be exponentially distributed random variables with parameter $1$ and let $U=\operatorname{min}\{X,Y\}$ and $V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$. We ...