Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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32
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8answers
3k views

The Monty Hall problem

I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the ...
32
votes
7answers
5k views

Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length 1 in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in 1 or 2 dimensions returns to the origin ...
20
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1answer
3k views

Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
16
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3answers
1k views

Card doubling paradox

Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value $x$) and after ...
14
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6answers
5k views

Zero probability and impossibility

I read a comment under this question: There are plenty of events that can occur that have zero probability. This reminds me that I have seen similar saying before elsewhere, and have never ...
104
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7answers
6k views

Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ coming from ...
18
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9answers
3k views

Reference book on measure theory

I post this question with some personal specifications. I hope it does not overlap with old posted questions. Recently I strongly feel that I have to review the knowledge of measure theory for the ...
11
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2answers
974 views

How to split an integral exactly in two parts

This question is a by-product of a conversation with Theo Buehler in comments to this answer. Let's settle definitions. Definition Let $(\Omega, \mathcal{F}, \mu)$ be a measure space. We say that ...
9
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2answers
1k views

What is an example of a lambda-system that is not a sigma algebra?

What is an example of a lambda-system that is not a sigma algebra?
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4answers
314 views

Probability Problem on Divisibility of Sum by 3

From the 3-element subsets of $\{1, 2, 3, \ldots , 100\}$ (the set of the first 100 positive integers), a subset $(x, y, z)$ is picked randomly. What is the probability that $x + y + z$ is divisible ...
10
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1answer
650 views

Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly

While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel ...
4
votes
1answer
652 views

how to show convergence in probability imply convergence a.s. in this case?

Assume that $X_1,\cdots,X_n$ are independent r.v., not necessarily iid, Let $S_n=X_1+\cdots+X_n$, suppose that $S_n$ converges in probability, how can we show that $S_n$ converges a.s.?
12
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2answers
1k views

Uniqueness of product measure (non $\sigma$-finite case)

Let $(X,\mathscr{A},\mu), (Y,\mathscr{B},\nu)$ be two measure spaces, then we have the product measurable space $(X\times Y, \mathscr{A}\times\mathscr{B})$ where $\mathscr{A}\times\mathscr{B}$ is the ...
3
votes
2answers
538 views

Combinatorics Distribution - Number of integer solutions Concept Explanation

I reading my textbook and I don't understand the concept of distributions or number of solutions to an equation. It's explained that this problem is 1/4 types of sampling/distributions problems. An ...
4
votes
1answer
434 views

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
3
votes
2answers
633 views

Tightness condition in the case of normally distributed random variables

Let $(X_n)_{n\geq 1}$ be a sequence of random variables such that $X_n\sim N(\mu_n,\sigma_n)$ for all $n\geq 1$. Then i'm trying to deduce that if $(X_n)_{n\geq 1}$ is tight in the sense that $$ ...
24
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5answers
3k views

Intuition behind Conditional Expectation

I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, ...
7
votes
5answers
4k views

Example where union of increasing sigma algebras is not a sigma algebra

If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra? It seems closed under complement since for all ...
6
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6answers
2k views

Best measure theoretic probability theory book?

I'm looking for a clear way to learn measure theoretic probability theory. Any suggestions?
10
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2answers
1k views

Algebra of Random Variables?

I've been looking online (and in teaching journals) for a good introduction to Algebras of Random Variables (on an undergraduate level) and their usage, and have come up short. I know I can find the ...
12
votes
3answers
910 views

Is there a possibility to choose fairly from three items when every choice can only have 2 options

Me and my wife are often not knowing which DVD to watch. If we have two options we have a simple solution, I put one DVD in one hand behind my back and the other DVD in the other hand. She will ...
6
votes
1answer
403 views

Limit of sums of iid random variables which are not square-integrable

The Central Limit Theorem tells us that for an iid sequence of random variables $(X_n)_{n\geq 0}$ of finite variance $\sigma^2$ and zero mean $$\lim_{n\to\infty}\frac{S_n}{\sqrt{n}}=^d ...
4
votes
2answers
173 views

how to derive the mean and variance of a Gaussian Random variable?

How do we go about deriving the values of mean and variance of a Gaussian Ransom Variable $X$ given its probability density function ?
4
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3answers
2k views

Convergence of random variables in probability but not almost surely.

Can somebody provide me with a sequence of (real-valued) functions, say on $[0,1]$ with the Lebesgue measure, such that the sequence converges in probability, or maybe in $\Vert \cdot \Vert _{L^2}$, ...
1
vote
2answers
191 views

Find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables.

How do I find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables. I know X~U[0,1], -ln(x) is exponential(1). I also know the sum of two or more independent ...
1
vote
1answer
176 views

Markov processes driven by the noise

Let $\xi_n\in \Xi$ be a sequence of iid random variables with $n \in\mathbb N\cup\{0\}$, which we call a noise process. Construct a process $$ Z_{n+1} = f(Z_n,\xi_n)\quad(\star) $$ with $Z_0\in E$ ...
3
votes
2answers
296 views

Using Recursion to Solve Coupon Collector

I read a brilliant answer by Mike Spivey on one of the questions and I was wondering how I could use it to solve a coupon collectors problem. The problem is : There are coupons labelled 1,2,3...,10 ...
3
votes
2answers
570 views

Conditional expectation for a sum of iid random variables: $E(\xi|\xi+\eta)=E(\eta|\xi+\eta)=\frac{\xi+\eta}{2}$

I don't really know how to start proving this question. Let $\xi$ and $\eta$ be independent, identically distributed random variables with $E(|\xi|)$ finite. Show that ...
2
votes
1answer
424 views

Alternative Expected Value Proof

I am currently tasked with proving an alternative definition of the expected value function. Considering X to be a random variable that takes all positive integers, I have to prove that ...
1
vote
1answer
191 views

$X_n \stackrel{d}{\to} X$, $c_n \to c$ $\implies c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

Let $X_n$, $X$ random variables on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and $(c_n)_n \subseteq \mathbb{R}$, $c \in \mathbb{R}$ such that $c_n \to c$ and $X_n \stackrel{d}{\to} X$. ...
1
vote
1answer
209 views

When random walk is upper unbounded

Consider a random walk $S_n = a_1+\dots+a_n$ where $a_n$ are iid random variables with $Ea_1 = a$ and $E|a_1|<\infty$. I am interested in the case when $\sup\limits_n S_n>M$ for all $M$ a.s. ...
0
votes
1answer
156 views

Convergence in probability and almost surely

Let $X_n$ be a sequence of independent random variable which converges in probability to $X$. Prove $X$ is a constant. Can someone give me a hint how I should go about proving this? I tried proving ...
23
votes
8answers
4k views

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
32
votes
4answers
1k views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
12
votes
1answer
1k views

What is meant by a continuous-time white noise process?

What is meant by a continuous-time white noise process? In a discussion following a question a few months ago, I stated that as an engineer, I am used to thinking of a continuous-time ...
19
votes
1answer
721 views

How far can probability intransitivity be stretched?

Once upon a time I read about nontransitive dice - sets of dice where "is more likely to roll a higher number than" is not a transitive relation. After the surprise wore off, I wondered - just how far ...
8
votes
4answers
2k views

Intuitive explanation of variance and moment in Probability

While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment. What is a good way to think of those two terms?
7
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5answers
3k views

Intuitive explanation of the tower property of conditional expectation

I understand how to define conditional expectation and how to prove that it exists. Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, ...
10
votes
4answers
557 views

Probability of having zero determinant

Given a matrix $A_{n \times n}$, which has elements $a_{i,j} \sim \mathrm{unif} \left[a,b\right]$, what is the probablity of $\det(A)$ being zero? What if $a_{i,j}$ have any other distribution? ...
5
votes
1answer
394 views

For symmetric stable distributions, why is $\alpha \le 2$?

I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact. Suppose we are trying to come up with stable distributions. From the definition, ...
6
votes
2answers
2k views

Meaning of non-existence of expectation?

When reading another post, I was wondering about the definition of existence of expectation of a random variable. From Kai Lai Chung, We say a random variable $X$ has a finite or infinite ...
7
votes
1answer
899 views

The limit of a convergent Gaussian random variable sequence is still a Gaussian random variable

I'm trying to prove this conclusion but have some problems with one of the steps. Assume $X_1,\ldots,X_n,\ldots$ is a sequence of Gaussian random variables, converging almost surely to $X$, prove ...
6
votes
1answer
599 views

Convergence in law and uniformly integrability

I'm looking for an elementary way of showing the following. If $(X_n)$ and $X$ are random variables such that $X_n \to X$ in distribution and such that $\{X_n\mid n\geq 1\}$ are uniformly integrable, ...
5
votes
1answer
916 views

Another question on almost sure and convergence in probability

Convergence in probability implies convergence on a subsequence almost surely. But this means we fix a subsequence, such that $X_{n_k}$ converges for almost every $\omega$, right? The subsequence we ...
5
votes
2answers
622 views

Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
3
votes
2answers
1k views

Why does the median minimize $E(|X-c|)$?

Suppose $X$ is a real-valued random variable. Let $P$ be the probability measure of $X$. Then $$ E(|X-c|) = \int_\mathbb{R} |x-c| dP(x). $$ Its median is defined as a number $m \in \mathbb{R}$ ...
2
votes
2answers
159 views

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ random variables. What is the distribution of $X_1^2 + X_2^2$?

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
1
vote
1answer
188 views

Problems on expected value

I'm self studying probability theory and I'm stuck in the following problems 1) Prove the following for a random variable $X$ with cdf $F$ $$E(x)=\int_0^\infty (1-F(x)) dx - \int_\infty^0 F(x) dx$$ ...
0
votes
1answer
151 views

X and Y are i.i.d. X+Y and X-Y are independent. E(X)=0 and E(X^2)=1. Show that X~N(0,1)

$X$ and $Y$ are i.i.d. $X+Y$ and $X-Y$ are independent. $E(X)=0$ and $E(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic functions to prove this. Any ideas?
7
votes
4answers
254 views

How variance is defined?

The variance of a random variable $X$ is defined as $E[(x-\mu )^2]$. Why can't it be defined as $E[|x-\mu |]$. i.e., What is the basic idea behind this definition. Thank you.