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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1answer
441 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, ...
10
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4answers
638 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? ...
6
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2answers
3k 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 ...
5
votes
1answer
6k 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 ...
8
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2answers
818 views

Prokhorov metric vs. total variation norm

Let $(S,d)$ be a metric space and let $\mathcal P(S)$ denote the space of Borel probability measures on $S$ endowed with the Prokhorov metric $\pi:\mathcal P(S)\times \mathcal P(S)\to \mathbb R_+$ ...
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4answers
1k 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 ...
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1answer
1k 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 ...
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vote
1answer
197 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 $F$ $$E(x)=\int_0^\infty (1-F(x)) dx - \int_\infty^0 F(x) dx$$ ...
7
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1answer
1k views

Asymptotics of binomial coefficients and the entropy function

I found a question while I was trying to practice Combinatorics and Probabilistic methods.I tried to solve it with no success.. this is the question: Use the Stirling approximation of the ...
6
votes
1answer
703 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
2answers
802 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 ...
4
votes
1answer
801 views

Understanding the relationship of the $L^1$ norm to the total variation distance of probability measures, and the variance bound on it

I am trying to find a bound for variance of an arbitrary distribution $f_Y$ given a bound of a Kullback-Leiber divergence from a zero-mean Gaussian to $f_Y$, as I've explained in this related ...
3
votes
2answers
184 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, ...
7
votes
4answers
296 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.
6
votes
1answer
251 views

Can you make money on coin tosses when the odds is against you?

The strategy Given an initial investment $n$ dollars and a "bet buffer" $b$. Calculate the bet size $x=\lfloor\frac{n}{2^b-1}\rfloor$ dollars. Wager $x$ dollars on random variable $C$ that $C=1$ ...
4
votes
1answer
681 views

Weak convergences of measurable functions and of measures

My question is "how weak convergences of measurable functions is defined?" There seems to be two different definitions which are both based on weak convergence of measures generated by the measurable ...
4
votes
1answer
303 views

Feller continuity of the stochastic kernel

Given a metric space $X$ with a Borel sigma-algebra, the stochastic kernel $K(x,B)$ is such that $x\mapsto K(x,B)$ is a measurable function and a $B\mapsto K(x,B)$ is a probability measure on $X$ for ...
3
votes
2answers
10k views

Finding probability P(X<Y)

How can I find this probability $P(X<Y)$ ? knowing that X and Y are independent random variables.
3
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1answer
192 views

How expected value is related to density function?

Let $X$ be a random variable on $(\Omega, \Sigma, P)$. The expected value of $X$ is defined as $$EX = \int X \,dP.$$ But when we calculate $EX$, we often use $$ EX = \int_{-\infty}^\infty xf(x) dx ...
2
votes
2answers
740 views

Expected value of the minimum (discrete case)

Maybe related to this question In the comments of this question they say that it gets easier if the variables are identically and independently distributed. But i don't see how because in my case the ...
2
votes
1answer
753 views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
2
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5answers
183 views

Probability distribution functions for the perimeter and space of triangle with fixed radius

Given a circle with radius R = 1, I'm trying to find either the probability distribution function or the density function for the space of triangle, which is randomly selected on this circle. The same ...
2
votes
4answers
2k 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 ...
1
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1answer
103 views

polynomial approximation on compacts

Let's say $f:\mathbb{R}^d\rightarrow \mathbb{R}$ is of class $C^k$ with $k \geq 0$. How do I know that I can find a sequence of polynomials such that all its derivatives up to order $k$ converge ...
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1answer
474 views

Likelihood Function for the Uniform Density.

Let the random variable $X$ have a uniform density given by $$ f(x;\theta)=I_{[\theta-\frac{1}{2},\theta+\frac{1}{2}]} $$ where $-\infty\leq\theta\leq\infty $ the likelihood function for a sample of ...
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votes
2answers
274 views

Is first order moving average a Markov process?

Given first order moving average $$ x(n) = e(n) + ce(n-1) $$ where $e(n)$ is a sequence of Gaussian random variables with zero mean and unit variance which are independent of each other, and $c$ is ...
20
votes
3answers
21k views

Probability density function vs. probability mass function

I've an confession to make. I've been using pdf's and pmf's without actually knowing what they are. The idea that I've been having so long is that density = area under the curve but if I look at it ...
28
votes
3answers
2k views

Why is this coin-flipping probability problem unsolved?

You play a game flipping a fair coin. You may stop after any trial, at which point you are paid in dollars the percentage of heads flipped. So if on the first trial you flip a head, you should stop ...
6
votes
1answer
2k views

Interpretation of sigma algebra

My question is how to interpret sigma algebra, especially in the context of probability theory (stochastic processes included). I would like to know if there is some clear and general way to interpret ...
4
votes
1answer
469 views

Martingale and bounded stopping time

A theorem of submartingale and bounded stopping time says: Theorem 5.4.1. If $X_n$ is a submartingale and $N$ is a stopping time with $\mathbb P (N \le k) = 1$ then $\mathbb EX_0 ≤ \mathbb EX_N ≤ ...
12
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1answer
650 views

Meaning of “kernel”

In analysis, there are at least three kinds of "kernel" concepts: In probability theory, there is a concept called transition probability, also called probability kernel, from one measure space $X$ ...
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3answers
487 views

Finitely Additive not Countably Additive on $\Bbb N$

Does there exist a function defined on the power set of the natural numbers to the interval from $0$ to $1$, $p:2^{\Bbb N}\rightarrow [0,1]$, such that $p$ is finitely additive, i.e. ...
10
votes
3answers
362 views

Are polynomials dense in Gaussian Sobolev space?

Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of ...
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8answers
2k views

Applications of Probability Theory in pure mathematics

My (maybe wrong) impression is that while probability is widely used in science (for example, in statistical mechanics), it is rarely seen in pure mathematics. Which leads me to the question - Are ...
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2answers
160 views

What is probability? [closed]

I tried to understand the most fundamental foundation of the mathematical definition of probability in the most natural/human way. (At first, I thought I may have found a proper understanding like ...
3
votes
1answer
1k views

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} ...
5
votes
1answer
325 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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3answers
2k views

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 ...
3
votes
2answers
176 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 ...
2
votes
1answer
212 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 ...
1
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1answer
213 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 ...
9
votes
2answers
799 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 ...
9
votes
1answer
2k views

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 ...
8
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0answers
216 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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votes
4answers
10k views

Poisson Distribution of sum of two random independent variables $X$, $Y$

$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim ...
6
votes
0answers
171 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
8answers
4k views

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
2answers
213 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
105 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
944 views

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 ...