5
votes
1answer
84 views

What background is required to understand Random Matrix Theory

I would like to be able to understand RMT but after "reading" many articles I have found that I have a limited capacity. I just see complicated equations. I guess I know linear algebra, classical ...
0
votes
0answers
25 views

A measurement of “likelihood” of a roll of $n$ independent normal random variables

I'll start with a general question, and later, as I expect it is possible that the general case does not have a satisfying answer, I'll post my specific problem. I am trying to find some measurement ...
1
vote
1answer
74 views

Examples for Conditional Expectation (modern probability theory)

I'm in the process of learning about conditional expectation in the framework of modern probability theory. The sudden change brought about by the notion of conditional expectation being a function on ...
3
votes
1answer
72 views

Mathematics of a Simple Counting Game

I wonder how can one think mathematically about the following game: People sit in a circle. One of them says "One!". Then somebody (no matter who - he/she can even be the former person) says ...
3
votes
1answer
340 views

Real Analysis and Statistics

What level of real analysis do you think is desirable for the study of statistics? I know that for many statisticians with applied focus, rigorous mathematics tend to give them a headache and I am ...
8
votes
2answers
192 views

Why do we want probabilities to be *countably* additive?

In probability theory, it is (as far as I am aware) universal to equate "probability" with a probabilistic measure in the sense of measure theory (possibly a particularly well behaved measure, but ...
0
votes
2answers
92 views

Question on meaning probability of coin tosses?

I have a question on the fundamental meaning of probability. The most familiar example of probability, is the probability of $\frac 12$ for $\text{H}$ and $\text{T}$, each for a single toss of an ...
5
votes
1answer
135 views

Differentiation under (measure theoretical) integral sign

I am looking for a citable reference for the result on differentiation under the integral sign for integration against a measure. The result states that if $R \subset \mathbb R$, $(X,\mathcal F, ...
2
votes
2answers
207 views

Category for measure spaces?

I know some things about measures/probabilities and I know some things about categories. Shortly I realized that uptil now I have never encountered something as a category of measure spaces. It seems ...
1
vote
2answers
184 views

What is a real-valued random variable?

This question arose when someone (and surely not the least!) commented that something like $\left(X\mid Y=y\right)$ , i.e. $X$ under condition $Y=y$, where $X$ and $Y$ are real-valued random variables ...
4
votes
3answers
973 views

Improbable vs Impossible?

I was wondering how mathematics in general or any of its sub fields e.g.statistics, probability, define the words Improbable and Impossible. I get their English meaning, that something is impossible ...
2
votes
1answer
97 views

Convergence w.p. 1 vs convergence in probability: a “physical” example

I understand (proved) that convergence with probability one implies convergence in probability, and that the latter notion is indeed weaker; I've completed an exercise showing that a sequence of ...
2
votes
1answer
175 views

Why do we need continuous random variables since they can be approximated by discrete ones?

I do not understand the motivation of developing the theory of continuous random variables. Given simple discrete random variables, the continuous ones can be well approximated.
7
votes
2answers
254 views

Applications of information geometry to the natural sciences

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
3
votes
1answer
158 views

What would be surprising facts in various areas of mathematics?

This might be too broad question - but what would be the things that you consider as surpising facts and discoveries in various areas of mathematics? If this seems too broad, it is fine to stick ...
-1
votes
1answer
134 views

What does it mean for a random variable to describe an experiment?

I often hear the expression, random variable (or sequence of rd's) to describes an experiment (or sequence of experiments. But that sound totally unrigorous to me: So we're given a mapping $X:\Omega ...
1
vote
1answer
321 views

What is the relationship between variance and energy

I was speaking with someone today who told me that variance, in the sense of probability theory, is equivalent mathematically to energy in physics. Can anyone elaborate on this relationship?
4
votes
3answers
271 views

Why are half-open intervals $(a,b]$ “special” in probability theory?

I'm learning probability theory and I see the half-open intervals $(a,b]$ appear many times. One of theorems about Borel $\sigma$-algebra is that The Borel $\sigma$-algebra of ${\mathbb R}$ is ...
3
votes
1answer
143 views

Reference request for examples of probabilistic heuristics, help put some examples in a broader context.

I was thinking about how probability is used in heuristic arguments, an example being the argument that there are an infinite number of twin primes: the probability that $n$ is the first of two twin ...
7
votes
1answer
764 views

What distinguishes the Measure Theory and Probability Theory?

It is clear that the Theory of Probability works primarily with limited measures on measurable spaces. On the other hand there is a folklore that says that what distinguishes the Theory of ...
1
vote
1answer
255 views

Radical Applications of Algebraic Topology

Are there any radical applications of algebraic topology? For example, in probability theory we look at sample spaces. Suppose the sample space is a torus (for example). Would computing homology ...
1
vote
1answer
73 views

Monty hosting a new show

I imagine the following setup. There is a contestant who has to pick one of three doors. How many prizes will be hidden is determined at random in the following way. Monty will toss a fair coin and ...
4
votes
5answers
419 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 ...
41
votes
5answers
4k 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 ...
21
votes
1answer
731 views

Expository articles on Analysis and Probability theory

When I come across a notion from algebra or number theory which I don't know I usually check Keith Conrad's page to see if he has written something about it. Key features of his articles are a very ...
2
votes
0answers
384 views

Relationship between Abstract Algebra and Probability Theory

Is there a relationship between abstract algebra and probability theory? I ask this because of the following laws: Axiom of Countable Additivity: If $A_1, A_2, \dots \in \mathcal{B}$ (where ...
2
votes
2answers
148 views

Testing what probability model to use

Suppose we have some random variable $X$ that ranges over some sample space $S$. We also have two probability models $F$ and $G$. Let $f(x)$ and $g(x)$ be the probability density functions for these ...
2
votes
3answers
395 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 ...