Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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Calculating stationary distribution of markov chain

I am asked to compute the stationary distribution of the markov chain with state space $E=\{0\dots,n\}$ and transition matrix below: \begin{bmatrix} 0 & 1 \\ \frac{1}{n} ...
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3answers
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Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance

I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement: ...
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1answer
357 views

Right continuous version of a martingale

This is an exercise in chapter 2 of the book "Continuous Martingales and Brownian Motion" by Revuz and Yor: Consider the probability space $([0,1], \mathcal{B}([0,1]), dx)$, where $dx$ denotes ...
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1answer
278 views

central limit theorem for a product

Given $-1\leq x_i\leq 1$ identically distributed random variables for $i=1,2,\dots n$. What is the distribution function of their product? Is there a central limit theorem for products if $n$ is ...
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Discontinuity points of a Distribution function [duplicate]

Possible Duplicate: Distribution Functions of Measures and Countable Sets The question at hand is: Let F be a distribution function on $\mathbb{R}$. Prove that F has at most countably many ...
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1answer
493 views

Expectation of $TS_T$ where $T$ is the absorption time at $\{a,-a\}$ of a simple symmetric random walk $\{S_n\}$

I was trying to calculate the expectation of $T^2$ using some martingale and got that I needed the expectation of $TS_T$. Any idea?
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0answers
145 views

Independence in infinite sequence of random variables

I was reading this page from planetmath. It states the following, Consider an infinite sequence of independent random variables $\{X_n,n\in \mathbb{N}\}$. Using the Monotone Class Theorem one ...
3
votes
2answers
204 views

What tools are used to show a type of convergence is or is not topologizable?

There are many types of convergence. For example, in measure theory and probability theory, there are many types of convergence of measurable mappings (random variables). in measure theory and ...
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2answers
546 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
2
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1answer
233 views

Conditional expectation and martingales

I have a few questions concerning martingales. Let $Y\in \mathcal{L}^1(\Omega,\mathcal{F},\mathbb{P})$ be given, and $(\mathcal{F}_n)$ a filtration, and define $X_n:=\mathbb{E}[Y|\mathcal{F}_n]$. We ...
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vote
1answer
331 views

Borel-Cantelli lemma problem

Practice problem for exam: Let ${A_n}$ satisfy $\sum_{n=1}^\infty P(A_n \cap A^c_{n+1}) < \infty$ and $\lim_{n\to \infty} P(A_n) = 0$. Show that $P(\lim \sup A_n) = 0$. I can see that it is ...
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1answer
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Under what circumstance will a covariance matrix be positive semi-definite rather than positive definite?

I have a covariance matrix: $\operatorname{cov}(\mathbf{X}, \mathbf{X}) = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^T]$ According to ...
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votes
2answers
881 views

Completeness of a finite direct sum of closed subspaces of $L^2$

Let $X_1$ and $X_2$ be real-valued square-integrable random variables defined on a probability space $(\Omega, {\cal F},P)$. For $i=1,2$, set $$ A_i := \{g(X_i)\in L^2 \mid g \text{ is some Borel ...
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0answers
228 views

Uniqueness of the random variable from its distribution [closed]

Moderator's Note: This question has been put on hold due to the version over at MathOverflow having received better attention and produced an accepted answer. Interested readers are advised to ...
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3answers
244 views

Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?

It is frequently stated (in textbooks, on Wikipedia) that the "Law of large numbers" in mathematical probability theory is a statement about relative frequencies of occurrence of an event in a finite ...
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1answer
523 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
6
votes
1answer
513 views

stopped filtration = filtration generated by stopped process?

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
6
votes
1answer
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Is $p$-norm decreasing in $p$?

I could show that $\|\cdot\|_p$ is decreasing in $p$ for $p\in (0,\infty)$ in $\mathbb{R}^n$. Following are the details. Let $0<p<q$. We need to show that $\|x\|_p\ge \|x\|_q$, where $x\in ...
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8answers
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Expected value is a linear operator? Under what conditions is median also a linear operator?

I have always taken for granted that expected value is a linear operator. For any random variables $X$ and $Y$: $E(aX + bY) = aE(X) + bE(Y)$. Can anyone point me to a rigorous proof of this? Also, I ...
5
votes
0answers
162 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
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votes
2answers
243 views

Proof of the infinitude of primes by probabilistic methods.

I'm looking if there is proof of the infinitude of prime numbers using probabilistic method. I am motivated by the answer of my question here. The answer is based on a relationship between ...
5
votes
1answer
109 views

$X+Y\in L^1$ implies $X \in L^1$ given $X$ and $Y$ are independent random variables

This problem can be found here, which is a previous prelim exam problem of UT Austin. Let $X$ and $Y$ be two independent random variables with $X+Y \in L^1$. Show that $X\in L^1$. Generally, ...
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2answers
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Showing $\cos(t^2)$ is not a Characteristic Function

Usually when we try to show a function is not a characteristic function, we would prove it is not uniformly continuous. I am wondering if there is any other way to show $\cos(t^2)$ is not a ...
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1answer
198 views

Joint distribution by independent distributions

We have $N$ independent discrete finite random variables (RVs) $X_1,\dots,X_i,\dots,X_N$ where RV $X_i$ has $M_i$ finite number of elements. We are free to choose any distribution $f_i$ for RV $X_i$ ...
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2answers
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Explanation of Lyapunov condition of CLT

I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables: Lyapunov CLT. Let $s_n^2 = ...
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votes
2answers
390 views

Let $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. Find $EZ$.

Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$. Now I need to find $EZ$. Should I start like ...
4
votes
1answer
263 views

Is the set of all probability measures weak*-closed?

Let $(\Omega,\Sigma)$ be a measurable space. Denote by $ba(\Sigma)$ the set of all bounded and finitely additive measures on $(\Omega,\Sigma)$ (see http://en.wikipedia.org/wiki/Ba_space for a ...
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2answers
114 views

Conditions under which the Limit for “Measure $\to 0$” is $0$

Let $\mu$ be a probability measure on $X$, so that $\int_X \mu(dx) = 1$. Say under which conditions on the function $f: X \rightarrow \mathbb{R}_{> 0} \ $ (that is measurable and integrable) we ...
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6answers
497 views

Sleeping Mathematician (Sleeping Beauty)

I came across the following thought experiment, and I would like to understand whether the controversy around it is justified. Imagine an experiment in which a mathematician is put to sleep with some ...
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2answers
78 views

Conditional Expectation of Functions of Random Variables satisfying certain Properties

Suppose that we have a probability space $(\Omega, \mathcal{F}, P)$. Let $X,Y$ be real-valued random variables defined on this space, and let $\mathcal{H} \subset \mathcal{F}$ be a sub-sigma-algebra. ...
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1answer
88 views

Pairwise independence of Random variables does not imply indendence

Show by a counterexample that for a family $(X_i)_{i\in I}$ of random variables the independence of all pairs $(X_i,X_j)$ with $i,j\in I, i\neq j$ does not imply the independence of the family ...
3
votes
1answer
143 views

Functions and convergence in law

Let $X$ be a random variable taking values in some metric space $M$. Let $\{\phi_n\}$ be a sequence of measurable functions from $M$ to another metric space $\tilde M$. Suppose that $\phi_n(X)$ ...
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votes
2answers
968 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 ...
3
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1answer
302 views

a.s. Convergence and Convergence in Probability

Let $(\Omega, \mathcal A,\mathbb P)$ be such that $\Omega$ is countable and $\mathcal A = 2^{\Omega}$. Prove that almost sure convergence and convergence in probability are the same on this ...
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votes
2answers
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Transform uniform distribution to normal distribution using Lindeberg–Lévy CLT

Currently i am developing a game which is based on many computations of random values and therefore i have implemented many algorithms like the Mersenne-Twister etc. Unfortunately, all generators ...
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votes
2answers
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application of strong vs weak law of large numbers

By definition, the weak law states that for a specified large $n$, the average is likely to be near $\mu$. Thus, it leaves open the possibility that $|\bar{X_n}-\mu| \gt \eta$ happens an infinite ...
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3answers
453 views

Conditional mean on uncorrelated stochastic variable

I know that $E[X|Y]=E[X]$ if $X$ is independent of $Y$. I recently was made aware that it is true if only $\text{Cov}(X,Y)=0$. Would someone kindly either give a hint if it's easy, show me a reference ...
2
votes
2answers
141 views

I want to show $E(X)=\int_{0}^{\infty}P(X\ge x)dx$ for non-negative random variable $X$

Show that for a non-negative random variable $X$, $$\mathbb E(X)=\int_{0}^{\infty}\mathbb P(X\ge x)dx.$$ I started with $$\mathbb ...
2
votes
1answer
108 views

Cox derivation of the laws of probability

I have read Jaynes' Probability Theory: The Logic of Science a while ago, but mostly skimmed over parts of his derivations that I didn't immediately understand. Now I'm trying to really understand it, ...
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1answer
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Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent

If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent? What happens with the ...
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votes
1answer
120 views

A question about a stochastic process being Markov

Let $(X_{s},\mathcal{F}_{s})$ be a stochastic process adapted to a given filtration. I was told that, in order to prove that $X$ is Markov, it suffice to prove that for any nonnegative, ...
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5answers
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Some case when the central limit theorem fails

If I understand correctly, for various versions of the central limit theorems (CLT), when applying to a sequence of random variables, each random variable is required to have finite mean and finite ...
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1answer
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Rule with independent random variables and conditional expectations

I want to use a rule for conditional expectation I found in (German) wikipedia, not in my script/textbook of probability theory, I guess it should be simple and follow more or less straight from the ...
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0answers
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Expected value of max/min of random variables

I am trying to solve the following problem. Let there be $n$ urns and let us have $k$ balls. Assume we put every ball into one of the urns with uniform probability. Denote by $X_i$ the random ...
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1answer
72 views

Approximate normal distribution

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
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1answer
51 views

Conditional mean on uncorrelated stochastic variable 2

This question is a follow up from this. I was in doubt if to add it in my previous question, but thought it unfair to the great answers I had received. Let $X,Y$ be stochastic variables such that ...
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2answers
247 views

The distribution of barycentric coordinates

Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming ...
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2answers
2k views

What is linearity of Expectations?

In reading about the average case analysis of randomized quick sort I came across linearity of expectations of indicator random variable I know indicator random variable and expectation. What does ...
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1answer
520 views

Prove that vector has normal distribution

You are given two independent random variables: $W \sim \mathrm{Exp}(1)$, $Q \sim U([0; 2\pi ])$. Also, $a$ is a constant, chosen from $[-\pi/2; \pi/2]$. You build following random variables, based ...
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1answer
566 views

recursive equation for number of white balls

Consider a polyurn scheme of more than two colors. Let us draw a ball from the urn and replace it with another ball of the color we picked from the urn. We assume that $w$ is the number of white balls ...