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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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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2answers
408 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 ...
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1answer
282 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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115 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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525 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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1answer
96 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 ...
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1answer
154 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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2answers
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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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1answer
344 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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639 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 ...
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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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2answers
92 views

Interesting variant of binomial distribution.

Suppose i had $n$ Bernoulli trials with $X_{i}=1$ if the $i$th trial is a success and $X_{i}=-1$ if it is a failure each with probability $\frac{1}{2}$. Then the difference between the number of ...
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3answers
535 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 ...
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2answers
144 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 ...
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2answers
315 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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1answer
222 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 ...
2
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1answer
124 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
3k views

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 ...
2
votes
3answers
457 views

Uniform distribution on $\mathbb Z$ or $\mathbb R$

I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set - or even subsets of $\mathbb R$ of a ...
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3answers
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limit in probability is almost surely unique?

I read this proposition in a book, which was not proved. And I cannot verify it myself. Could anyone help me out here? If $$X_{n}\rightarrow X$$ in probability and $$X_{n}\rightarrow Y$$ almost ...
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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
73 views

What is the joint distribution of two random variables?

Today I was thinking about this and I have the feeling I am missing something obvious, but I can't seem to solve it. Suppose we have a continuos random variable $X$ with density $f_X(x)$. Let $Y = ...
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1answer
74 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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2answers
908 views

Conditional Probability and the Complement Rule

Does this identity hold for all events? $$ P(A|B) = 1-P(A'|B) $$ Logically speaking, if the probability of $A$ given $B$ occurred is $X$, shouldn't the probability that $A$ does not occur, $A'$, ...
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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
3k 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
527 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
609 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 ...
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1answer
78 views

One confusion over conditional expectation

Suppose for two random variables $X_1$ and $X_2$ $\sigma(X_1) = \sigma(X_2)$. Why $E[Y | X_1] = E[Y | X_2]$ a.e. ? the set where $X_1= X_1(\omega)$ can be different from the set $X_2= X_2(\omega) $. ...
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1answer
467 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim ...
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1answer
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What is the PDF of a product of a continuous random variable and a discrete random variable?

Let $N$ be a discrete random variable which takes values in [0, ..., M], M > 0, with known PDF $P(N=n)$. Let also the continuous random variable $Z = \sum_{i=1}^{N}X_{i}$ as the sum of i.i.d. $X_{i}$ ...
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1answer
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pdf equation for tossing 2 coins given the probability of landing head for each coin in a single toss

Problem: Consider a simple coin-flipping experiment in which we are given a pair of coins A and B of unknown biases, $\theta_{A}$ and $\theta_{B}$ respectively (that is, on any given flip, coin A ...
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803 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? ...
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1answer
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+100

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
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1answer
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The strong and weak laws of large numbers: Why two?

The following questions are entirely based on the corresponding article from Wikipedia. The assumptions of both laws are the same, and the strong law has a more general claim than the one of the weak ...
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2answers
197 views

$X$ and $f(X)$ independent $\Longleftrightarrow$ $f(X)$ is degenerate

Let $(\Omega, \cal{A}, \mathbb{P})$ be a probability space and $X$ a random variable on $\Omega$. Let, also, $f:\Omega\to\mathbb{R}$ be a Borel function. Then: $X$ and $f(X)$ are independent ...
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0answers
214 views

Regular Version of Conditional Gaussian Distribution

Let $Z_{1}$ and $Z_{2}$ be two independent normally distributed random variables with expectations $\mu_{1},\mu_{2}\in\mathbb{R}$ and variances $\sigma_{1}^2,\sigma_{2}^2\in (0,\infty)$ . I would ...
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0answers
181 views

Expected Number of Convex Layers and the expected size of a layer for different distributions

It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a ...
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2answers
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Dual space of the space of finite measures

Since I am reading some stuff about weak convergence of probability measures, I started to wonder what is the dual space of the space consisting of all the finite (signed) measures (which is well ...
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2answers
204 views

show that if $X\ge 0$ , $E(X)\le \sum_{n=0}^{\infty}P(X>n)$.

if $X$ is a random variable and also let $X\ge 0$ , I want to show $E(X)\le \sum_{n=0}^{\infty}P(X>n)$.
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1answer
849 views

Relations between Order Statistics of Uniform RVs and Exponential RVs

Say we have $U_1 \dots U_n$ i.i.d. random variables uniform on $[0,1]$ and $Y_1 \dots Y_{n+1}$ i.i.d. random variables distributed as $Y_i \sim Exp(1)$. I know that the joint distribution of the order ...
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2answers
58 views

How can I show that the “binary digit maps” $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?

In this post What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion? we needed the fact that if we let $b_i (x) \in \{0,1\}$ for ...
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2answers
136 views

$L^1$ norm of product of independent random variables

I am trying to show that $\|XY\|_1 = \|X\|_1\|Y\|_1$ for $X,Y$ independent random variables, where $\|X\|_1 = \int{|X| d\mathbb{P}}$. I have a feeling that this result is intuitive, but could anyone ...
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1answer
231 views

Applying Ergodic Theorem on fractional Brownian motion

For a fractional Brownian motion $B_H$ consider the sequence for $p>0$ $$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$ By the Ergodic Theorem it is ...
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1answer
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Why does a time-homogeneous Markov process possess the Markov property?

Klenke defines (Definition 17.3, p. 346) a time-homogeneous Markov process independently, rather than as a special case of a stochastic process that possesses the Markov property (Definition 17.1, p. ...
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2answers
2k views

Sample variance derivation

I have quite a simple question but I can't for the life of me figure it out. For a set of iid samples $\,\,X_1, X_2, \ldots, X_n\,\,$ from distribution with mean $\,\mu$. If you are given the sample ...
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370 views

Inequalities involving the probability density function and variance

I am wondering whether anyone knows of any any inequalities involving the probability density function of an unknown distribution (as opposed to the cumulative distribution function) and its known ...
4
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2answers
999 views

Connection to Normal distribution

I've been working on finding the probability for the event, that the sum of $n$ independent random variables are less than $s$, when they are evenly distributed on $[0,1)$. I've used the law of total ...
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1answer
94 views

Exercise on Martingales

I have been struggling with the following exercise and I was wondering whether my solution is correct or not. I am pretty sure about the second part of the question (the martingale part) but not so ...