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 [tag:probability-...

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17
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3answers
5k views

How to deduce the CDF of $W=I^2R$ from the PDFs of $I$ and $R$ independent

Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find cdf of $W =I^2R$? Where, $$ \begin{align} f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\ f_R(r)&=2r, &0 \leq r\...
48
votes
10answers
7k 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 ...
29
votes
6answers
17k 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 ...
34
votes
1answer
6k 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, ...
10
votes
1answer
2k 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 $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
44
votes
7answers
12k 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 ...
12
votes
1answer
3k 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 ...
180
votes
8answers
11k 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 ...
34
votes
13answers
9k 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 ...
21
votes
3answers
2k 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 ...
15
votes
3answers
9k 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 $...
16
votes
3answers
8k views

Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not ...
7
votes
1answer
23k views

Gamma Distribution out of sum of exponential random variables

I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density ...
15
votes
2answers
919 views

If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely

Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely. The hint I was given is to evaluate: $$E[X-Y;X>a,...
15
votes
3answers
8k views

Are functions of independent variables also independent?

It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed. If I have two independent random variables, $X_1$ and $X_2$, then I define two ...
16
votes
1answer
2k 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 $\sigma$-...
3
votes
1answer
138 views

When Superposition of Two Renewal Processes is another Renewal Process?

When superposition of two renewal processes is another renewal process? If you merge (superpose) two Poisson processes with parameters $\lambda_1$ and $\lambda_2$, the outcome is another Poisson ...
6
votes
1answer
1k 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.?
13
votes
1answer
2k views

Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $ (\Omega, \mathscr{A}, \mu ) $ ...
5
votes
4answers
2k 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 ...
11
votes
2answers
1k 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 $X\...
8
votes
4answers
4k views

Question on the 'Hat check' problem

The famous 'Hat Check Problem' goes like this, 'n' men enter the restaurant and put their hats at the reception. Each man gets a random hat back when going back after having dinner. The goal is to ...
3
votes
3answers
4k views

biased random walk on line

Lets say we start at point 1. Each successive point you have a, say, 2/3 chance of increasing your position by 1 and a 1/3 chance of decreasing your position by 1. The walk ends when you reach 0. ...
2
votes
2answers
243 views

Show that $\prod (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$

Let $A_n$ be independent events with $P(A_n) \neq 1$. Show that $\prod_{n=1}^{\infty} (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$ It kind of looks obvious but I really have no idea how to prove it. Can ...
47
votes
5answers
6k 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, ...
43
votes
8answers
10k 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 ...
59
votes
6answers
9k views

Chance of meeting in a bar

Two people have to spend exactly 15 consecutive minutes in a bar on a given day, between 12:00 and 13:00. Assuming uniform arrival times, what is the probability they will meet? I am mainly ...
13
votes
2answers
3k 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?
8
votes
4answers
4k views

Why isn't there a uniform probability distribution over the positive real numbers?

Apparently, the solution to the Card Doubling Paradox is that a uniform probability distribution over the positive real numbers doesn't exist. Can anyone explain why this is the case and what ...
4
votes
2answers
3k views

Transformation of state-space that preserves Markov property

I am solving a problem in Mathematical Statistics by Jun Shao Let $\{X_n \}$ be a Markov chain. Show that if $g$ is a one-to-one Borel function, then $\{g(X_n )\}$ is also a Markov chain. Give an ...
9
votes
3answers
5k 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
1answer
33 views

Thinning a Renewal Process - Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a ...
3
votes
1answer
403 views

If $X_n \stackrel{d}{\to} X$ and $c_n \to c$, then $c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

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
208 views

What is the name of this theorem, and are there any caveats?

For random variable $X$ that follows some distribution, $f(x)$ is the probability density function of that distribution if and only if $$\mathbb{E}[\phi(X)] = \int_{-\infty}^\infty \phi(x) f(x)dx$$ ...
1
vote
2answers
239 views

Prove X is a martingale

Prove $X = (X_n)_{n \geq 0}$ is a martingale w/rt $\mathscr{F}$ where X is given by: $X_0 = 1$ and for $n \geq 1$ $X_{n+1} = 2X_n$ w/ prob 1/2 $X_{n+1} = 0$ w/ prob 1/2 and $\mathscr{F_n} = \...
36
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 ...
29
votes
3answers
2k views

Random Variable Inequality

Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result: (A-S 1....
15
votes
2answers
1k views

Conditional expectation on more than one sigma-algebra

I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two sigma-...
14
votes
2answers
2k 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 $...
11
votes
2answers
1k views

Measurability of one Random Variable with respect to Another

After several hours of struggling, I've been unable to solve the following problem Let $X,Y: (\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{R})$ where $\mathcal{R}$ are the Borel Sets for ...
8
votes
2answers
778 views

Conditional expectation equals random variable almost sure

Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$. Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely. I ...
9
votes
3answers
999 views

Difference between Modification and Indistinguishable

Would someone be able to offer a layman's explanation of what is means when two stochastic processes are a Modification of each other and when they are Indistinguishable? My Stochastic Analysis notes ...
3
votes
2answers
116 views

Correlated joint normal distribution: calculating a probability

Given $$ f_{XY}(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}} \exp \left( -\frac{x^2 +y^2 - 2\rho xy}{2(1-\rho^2)} \right) $$ $Y = Z\sqrt{1-\rho^2} + \rho X$ And $$ f_{XZ}(x,z) = \frac{1}{2\pi } \exp \...
4
votes
2answers
1k 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 ...
2
votes
2answers
271 views

Show that $E(X)=\int_{0}^{\infty}P(X\ge x)dx$ 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 E(X)=\int_{0}^{\infty}xdF(x)=\int_{0}^{\infty}\int_{0}^{x}dtdF(x)....
1
vote
1answer
607 views

$X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0 $and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$

$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic ...
6
votes
3answers
1k views

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

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 $E(\xi\mid\xi+\eta)=E(\eta\mid\...
5
votes
2answers
503 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 ...
4
votes
2answers
2k 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 ...
4
votes
2answers
251 views

Is it correct to say that ($\color{red}{(} \limsup |W_k|/k\color{red}{)} \le 1) \supseteq \limsup \color{red}{(}|W_k|/k \le 1\color{red}{)}$?

Let $W_0, W_1, W_2, \dots$ be random variables on a probability space $(\Omega, \mathscr{F}, \mathbb{P})$ where $$\sum_{k=0}^{\infty}P(|W_k|>k) <\infty$$ Prove that $$\limsup \frac{|...