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 ...

learn more… | top users | synonyms (1)

3
votes
1answer
333 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 ...
3
votes
2answers
3k views

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 ...
2
votes
2answers
88 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 ...
2
votes
3answers
508 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
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 ...
2
votes
2answers
295 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 ...
2
votes
1answer
196 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, ...
2
votes
1answer
128 views

Probability that a sequence of random variables converges to 0 or 1

Fix $\alpha , \beta > 0$ with $\alpha + \beta = 1$ and consider the sequence of random variables defined by $X_0 = \theta \in (0,1)$ and $$P(X_n = \alpha + \beta X_{n-1} \mid X_{n-1} , ... , X_0) ...
2
votes
1answer
2k views

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
votes
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, ...
2
votes
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
455 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 ...
2
votes
1answer
1k views

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 ...
2
votes
3answers
1k views

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 ...
2
votes
0answers
2k views

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 ...
1
vote
1answer
67 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 = ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
1answer
602 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 ...
0
votes
1answer
76 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) $. ...
0
votes
1answer
454 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 ...
0
votes
1answer
1k views

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. $X_{i}$ ...
0
votes
1answer
2k views

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 ...
9
votes
2answers
765 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? ...
6
votes
1answer
1k views

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 ...
6
votes
2answers
196 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 ...
6
votes
0answers
211 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 ...
6
votes
0answers
179 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 ...
6
votes
2answers
1k views

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 ...
5
votes
2answers
202 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 $E(X)\le \sum_{n=0}^{\infty}P(X>n)$.
5
votes
1answer
841 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 ...
4
votes
2answers
126 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 ...
4
votes
1answer
225 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 ...
4
votes
1answer
1k views

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. ...
4
votes
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 ...
4
votes
0answers
367 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
votes
2answers
982 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 ...
3
votes
1answer
92 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 ...
3
votes
2answers
105 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
3
votes
1answer
77 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
3
votes
0answers
108 views

Random $0-1$ matrices

I'm working my way through the Oxford notes in Probabilistic Combinatorics and came across this question in one of the question sheets; I'd like to stress that this is not my homework: I'm simply ...
3
votes
2answers
641 views

How can I show that the conditional expectation $E(X|X)=X$?

I tried to show that $E(X|X=x)=x$, which would lead me to get $E(X|X)=X$ but I am having trouble doing so. I know that the definition of conditional expectation (continuous case) is: ...
3
votes
1answer
134 views

How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space?

I not solve the follow limit $$\lim_{p\rightarrow 0} \bigg[\int_{\Omega} |f|^p d\mu \bigg]^{1/p} = \exp\bigg[ \int_{\Omega} \log|f|d\mu \bigg],$$ where $(\Omega, \mathcal{F}, \mu)$ is a probability ...
3
votes
1answer
437 views

Bus stop probability question

People arrive at random times and independently at a bus stop and wait for the bus to arrive. The bus arrives at this stop once every hour. Thus, the waiting times of the people follow a uniform ...
3
votes
1answer
130 views

Product Measures

Consider the case $\Omega = \mathbb R^6 , F= B(\mathbb R^6)$ Then the projections $\ X_i(\omega) = x_i ,[ \omega=(x_1,x_2,\ldots,x_6) \in \Omega $ are random variables $i=1,\ldots,6$. Fix $\ S_n = ...
3
votes
4answers
197 views

Conjunction fallacy [duplicate]

I was reading this article which has the following question, Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with ...
3
votes
2answers
385 views

Expectation of a stopping time uniquely determined by a function

Let $(X_t)_{t\ge0}$ be a Markov chain on a finite state space $\Omega$, with transition probability $P$. Let $T$ be a stopping time such that $T=\min \{t\ge 0;X_t \in A \subset \Omega \}$.  If ...
3
votes
1answer
179 views

Relation: pairwise and mutually

Suppose we can define a relation $R$ over the sets $X_1, …, X_k$ for any natural number $k$, note not specified for a particular $k$. I was wondering if there is some definition or conditions ...