An Introduction to Stochastic Calculus
https://bjlkeng.io/posts/an-introduction-to-stochastic-calculus/
2025 is looking like a great year for work at the intersection of #categorytheory, #systemstheory, #controltheory, #machinelearning and #probabilitytheory, this thread will be a very biased collection of works (in no specific order) I'm hoping to read as soon as possible!
Starting with:
"Logical Aspects of Virtual Double Categories"
https://mastoxiv.page/@arXiv_mathCT_bot/113921693949589956
Given that I am surrounded by Mathematicians here, let me ask for help for what should be a simple problem I can't seem to be able to solve:
Assume you have n fair dice with m faces (i.e. each can roll an integer from 1 to m with a uniform probability). You roll all n, and keep the k (with 0<k<=n) highest results. What is the probability that the sum of the k dice you kept is X?
(If one keeps all the dice, probability-generating functions give the answer straightforwardly. If I roll 2 dice and keep 1 I can easily enumerate the outcomes and calculate the probabilities, but I am stumped by the general case).
Fun little exercise: prove that if two Bernoulli random variables are perfectly correlated then they must be equal. #probabilitytheory
#FinishedReading #Boole 's Laws of Thought, whose final chapters partly shift attention from #logic to #probabilityTheory and #philosophyOfScience . I don't have a strong sense of his historical standing in either discipline, although for probability there is https://en.m.wikipedia.org/wiki/Boole%27s_inequality . The attraction of probability is clear, with its range of values from 0 to 1, use of '1 -' for negation, multiplication for conjunction (of independent events) etc. There are rhymes here with his logic at least
There are many situations in the real world where small initial differences can easily grow into very large differences just out of pure chance.
Since we are on a social network, let's create a toy model* where a number of posts all have the same probability to be reposted/shared/boosted by any person seeing them. Since the more people see a post, the more people have a chance of boosting it, the posts with more visibility are also the ones that are likely to gain more visibility. So small initial fluctuations (just one or two extra boosts at the beginning) can lead a post to skyrocket in popularity, even though it is not intrinsically "better" than any of the other.
If we simulate this process numerically and make a histogram of the result, we see that the distribution of how many boosts a post had rapidly grows a tail, with most posts having no visibility whatsoever, and a few having a LOT more than the average.
#ITeachPhysics #ProbabilityTheory #ToyModel
* In the #Physics jargon, a "toy model" is a very simple (often unrealistic) model, which nevertheless capture the essence of the problem, without being burdened by all the real world complications. If you ever heard about spherical cows in vacuum, that is a toy model!
While waiting for a meeting to start, I decided to revisit one of the classics
#graphicalmodels #probabilityandstatistics #probabilitytheory #probability #DAG
Now (7pm ET Wed) watch https://youtu.be/QS0VmCD9YLU(FEEL FREE TO SUBSCRIBE TO YOUTUBE
@hajiaghayi
FOR FUTURE LESSONS) Lesson 14: Introduction to Algorithms by Mohammad Hajiaghayi: We talk about #Probability (Part 2) useful for designing #randomized #algorithms
#algorithms, #design, #induction, #recursive, #randomizedalgorithms, #probability, #randominput ,
#probabilitytheory, #randomvariables, #expectations, #variance, #Bernoulli, #Binomial, #Poisson, #Normaldistribution, #Gaussian, #Python, #numpy.random, #scipy.stats, #correlation, #Pearson, #spearman, #geeksforgeeks , #hackerrank, #leetcode, #cs, #computerscience
@pgrepds I thought I coudn't understand #probability.
Thanks to a teacher, I understand better the relationship between #probabilitytheory, #settheory and #topology. Now, I think I will be able to work these different fields together.
I get an extra layer of humor here by being amongst the LessWrong crowd...
#Introduction time!
I'm Lika, a data analyst. I have an organic chemistry background.
Hope to join an academic program focused on modern aspects of #combinatorics in the near future.
#highereducation #linearalgebra #machinelearning #mathteaching #neuralnetworks #probabilitytheory #statistics #documentaryfilms #boardgames #rocksinging
Fun fact: This year I had experience as a museum tour guide. I was telling about Andrei Sakharov: his study and research in the field of theoretical physics.
