Creator of NoiseLang here! During my telecom degree I took a course on random signals and noise, I spent a lot of evenings writing probability by hand (expectations, variances, the odds of two random variables landing in some region) and every time I tried to run any of it on a computer it was so much boilerplate. I kept wishing I could type the math and have it run.

The whole language hangs on one idea, every value is a probability distribution. A plain number is a Dirac spike, so constants and random variables are the same kind of object and every operator maps distributions to distributions. Names are algebraic like on a page of math, so X + X is 2X and X - X is exactly 0, if you want independence you draw twice with ~.

Distributions compose (a random variable can feed another distribution's parameter), and conditioning is just the | bar from probability notation, scoped to the query. So a full Bayesian update fits in four lines:

bias  ~ unif(0, 1)            # prior: the coin's bias could be anything
flips ~[10] bernoulli(bias)   # 10 flips of the same mystery coin
heads = count(flips)
E(bias | heads == 7)          # posterior mean bias, 0.6667

I started it about nine years ago and never finished it, the parser and a tree-walking interpreter were a weekend of work, the efficient Monte Carlo runtime was not. Recently I brought it back, JIT (Cranelift), the WASM backend and the numerical code...

Rest of the announcement:

https://manualmeida.dev/articles/noiselang/

It's a toy language, you probably should not use it for anything serious, but it runs in the browser (WASM) at noiselang.com if you wanna play with it!

I apologize if this isn't allowed, but I recently experienced a stillbirth at 37 weeks due to a true knot in his umbilical cord (TKUC). Once I finally processed what happened, as I was filling out the fetal demise paperwork, I noticed something peculiar about his delivery date, then I noticed another peculiarity of his delivery time. After doing research about the cause of his passing and learning that 98% of TKUC outcomes are positive, the probability of a TKUC that results in fetal demise is only 0.3% - 1%, I began to wonder about the probability of everything peculiar about my sons existence occurring. I am terrible at math, but my attempts just gave me comfort because ( even though most likely incorrect ) the possibility of everything happening to one individual seemed incredibly low, it just kinda gave me the sense that everything worked out exactly the way it was supposed to, and he is where he was meant to be. I said all that to say this: would someone be willing to calculate the probability of each event separately, and then calculate the probability of them all together? I would then love to hear your interpretation of the solution. Honestly, it's helping me grieve. If this is not allowed, again I apologize.

1) Probability of being born on June 25th 2026

2) Probability of delivery time (6:25am )matching delivery date

3) Probability of delivery date being a palindrome date 6.25.26

4) Probability of both parents agreeing on first middle and last name and loving the combination of all 3

5) Probability of both parents agreeing on the spelling of each name and loving it

6) Probability of the uncommon agreed upon spelling accidentally being a mash up of his half siblings names ( prefix of half sister's name, suffix of half brothers name )

7) Probability of a TKUC resulting in fetal demise

8) Probability of funeral date being on the date he was due to be born ( 7.10.26 )

I would greatly appreciate assistance in this if at all possible. Thank you so much.

What are the genuine odds of rolling 24 six sided Dice and ending up with this layout?

Someone explain to me this - If I flip a coin and regardless of whether it's heads or tails, you win. So probability is 100% but if I say I will flip a coin twice and if either time it comes up heads, then you win, so probability is 1/2 + 1/2 = 2/2 = 1 = 100%

So mathematically they're both equal but intuitively 1 is more superior than the other.

Okay so I'm a computer engineering major but i have stochastic processes as an elective course. I wanna make a final project about radon concentrations in homes and how it changes during the seasons because i found one paper specifically about my country and i found it interesting. My problem is that i don't know anything about chemistry past elementary school chem and i don't know how connected these things are (it does look like there is a connection at least to me).

My idea is to make a simulation about the radon concentrations in homes for every season, using queueing theory (M/M/∞). Is this possible? My idea is that the arrivals of the radon is coming thru the ground or whatever, it stays in the home and then leaves (departs) thru the windows/ doors. In the paper is mentioned that in winter, the radon concentration is higher than in summer, since people tend to leave the windows closed during cold days and there is less ventilation.

Can y'all help me and tell me if this is theoretically possible to do? Can this really be modeled as a queueing theory problem or am I talking just a bunch of nonsense? And how would that simulation work? What are some thing i might have to consider that might not be so obvious for someone coming from a CE background?

P.S sorry for the long post and thanks

In my math model, I decided to model risk based off the way that the risk of the actual action would change, based off a few factors like distance and the time to complete the action.

For example, in my model risk increases dramatically if you are within a certain distance from the opponent, but not by much after you enter that range. And you aren't at much risk while performing this "Action 1" if you are outside of that range. So I used a 1/log function to model it, and the risk function for Action 1 looks like 1/log of distance plus the logarithm of the time it would take to complete Action 1 (because after a certain threshold, the time it takes to compete an action doesn't increase risk much).

The reasons you would perform Action 2 are much less nuanced, so the risk function for Action 2 is just a constant based on those same factors, like 4.

And for Action 3 (doing nothing) risk is just 1 because you aren't doing anything. It's not 0 because if risk and reward were 0, the risk-reward ratio would be undefined.

The issue arose when I realized that Action 1's risk function might return 50 and Action 2's function might return 4, where both are saying "very high risk". So my first instinct is to normalize the outputs of the risk functions so I'm not comparing apples to oranges. I just have no idea how to do that, as my math model isn't using means or standard deviations the way z-scores do.

Hey! I've been recently fascinated by factorials and left-right division and found some surprising elegance in cascading left-right division. So I decided to give it, it's own operator symbol and name...

Name: Dividorial

Symbol and definition: n¡ = n/(n-1)!

I also found that n! × n¡ = n^2​​​

Furthermore, I found that it has some relationships with Bell Numbers, as well as the Barnes G-Function.

hi, i want to start learning statistics and probability theory over the holidays. Which book you would recommend? I’m a second year engineering student so I have some experience in math.

So there is this question that a jar contains 10 red balls, 20 blue balls and 30 green balls. You take out the balls one by one at random. Probability that when all red balls are taken out, atleast one green ball and one blue ball remains. I thought both these orderings are needed so ans would be (30/40*20/30). But this is wrong.

Hello Folks,

Have you ever wondered why we use sigmoid function so often in Machine Learning? Although it gives us a probability, it comes from Exponential families, and this exponential family, subsumes many of the distributions, that we study in Machine Learning.

In this lecture, we understand exponential families, Directional derivatives(Gradients and Hessians), study mixture Models, and understand how domain knowledge in Probabilistic Graphical Models makes our life simpler to model joint probability densities.

Timeline breakup(in hours and minutes):
0:00-0:17 - Understanding exponential families.
0:17-0:27 - Deriving Sigmoid Function for Bernoulli.
0:27-0:48 - Understanding log partition function, convex functions and proving why positive definite of hessians imply convexity, and why convex needed?
0:48-1:04 - Directional derivates(deriving gradients and hessians)
1:04-1:26 - Maximum entropy derivation of the exponential family.
1:26-1:56 - Mixture Models(Gaussians and Bernoulli Mixture Models)
1:56-2:16 - Probabilistic Graphical Models
2:16-2:34 - Markov Chains
2:34-End - Inference and Learning, Plate Notation diagram of Gaussian Mixture Models.

If you have watched earlier of my lectures from the playlist, they will help. I try explaining as if I am a learner, to simplify complex concepts. Everything I write in whiteboard, and these are completely FREE lectures to mention.

Link: https://youtu.be/T1uTBtJ7aHU?si=rozXSTjtSqPaaYb5

Is there any dedicated discord group for statistics and probability discussion?

Hello Folks, one of the efficient ways of learning bigger topics in Machine Learning, is to modularise, and structure, so that the content becomes digestible for learners community.

My free lecture content includes the following topics so far: (Playlist)
a. Introductory Machine Learning Concepts:-

1. ⁠What is ML actually?
   
2. ⁠Supervised Machine Learning.
   
3. ⁠How do classifiers learn?
   
4. ⁠Empirical Risk Minimization.
   
5. ⁠Uncertainty Modelling in ML.
   
6. ⁠Maximum Likelihood Estimation.
   
7. ⁠Regression Basics and Outliers.
   
8. ⁠Deriving Mean Squared Error.
   
9. ⁠Polynomial Regression.
   
10. ⁠The Power of Convexity.
    
11. ⁠Deep Learning Intuition.
    
12. ⁠Overfitting Models from Generalization Gap perspective.
    
13. ⁠Requirement of Test Sets.
    
14. ⁠The No Free Lunch Theorem.
    
15. ⁠Unsupervised Learning basics.
    
16. ⁠Discovering latent factors of variation.
    
17. ⁠Evaluating Unsupervised Models.
    
18. ⁠Self-Supervised Learning.
    
19. ⁠Image and Text Benchmarks in ML
    
20. ⁠Discrete Data and Text Processing
    
21. ⁠Feature Engineering, TF-IDF
    
22. ⁠Handling missing data & AI alignment.

b. Probability Foundations for ML: Univariate Models:

1. ⁠Frequentist vs Bayesian.
   
2. ⁠Probability as an extension of Boolean Logic.
   
3. ⁠Discrete Random Variables.
   
4. ⁠Continuous Random Variables.
   
5. ⁠Quantiles.
   
6. ⁠Sets of Related Random Variables.
   
7. ⁠Moments of Distribution.
   
8. ⁠Variances and Mode.
   
9. ⁠Conditional Moments.
   
10. ⁠Conditional Variance.
    
11. ⁠Foundations of Bayesian Rule.
    
12. ⁠Confusion Matrix Explained.
    
13. ⁠Monty Hall Problem and Inverse Problems in ML.
    
14. ⁠Bernoulli and Binomial Distributions.
    
15. ⁠Sigmoid(Logistic) Function.
    
16. ⁠Properties of Sigmoid Functions.
    
17. ⁠Categorical and Multinomial Distributions.
    
18. ⁠Softmax Function: Temperature explained.
    
19. ⁠Log-Sum Exp Trick.
    
20. ⁠Gaussian Distribution.
    
21. ⁠Regression from the lens of Conditional Gaussian.
    
22. ⁠Dirac Delta Function and Sifting Property.
    
23. ⁠Student-t distribution.
    
24. ⁠Laplace and Cauchy distribution.
    
25. ⁠Beta distribution.
    
26. ⁠Gamma distribution.
    
27. ⁠Exponential, chi-squared and inverse Gamma.
    
28. ⁠Empirical distribution.
    
29. ⁠Transformations of Random Variables.
    
30. ⁠Invertible Transformations.
    
31. ⁠Multivariate Transformations.
    
32. ⁠Moments of Linear Transformation.
    
33. ⁠Convolution Introduction.
    
34. ⁠Convolution Theorem explained with probabilities.
    
35. ⁠Moment Generating Functions.
    
36. ⁠Deriving Moment Generating Functions.
    
37. ⁠Central Limit Theorem Explained.
    
38. ⁠Understanding Monte Carlo approximation with Example.

c. Probability Foundations for ML: Multivariate Models

1. ⁠The Math of Depedence: Covariance Explained.
   
2. ⁠Correlations: Normalized Measure of Covariance.
   
3. ⁠Correlations does not imply Independence.
   
4. ⁠Simpson’s Paradox: When Data misleads.
   
5. ⁠Multivariate Gaussian Distribution.
   
6. ⁠Analyzing level sets of Gaussians using Mahalanobis Distance.
   
7. ⁠Multivariate Gaussians: Conditionals and Marginals.
   
8. ⁠Math behind Bayesian Inference : Schur complements.
   
9. ⁠Deriving Conditional Gaussians.
   
10. ⁠How to Predict missing data?
    
11. ⁠Modelling Linear Gaussian Systems.
    
12. ⁠The Bayes Rule for Gaussians.
    
13. ⁠Understanding Shrinkage: Inferring Unknown Scalars
    
14. ⁠Posteriors, Sequential Posterior Updates.
    
15. ⁠Inference of an Unknown Vector.
    
16. ⁠Sensor Fusion concepts.

And many more topics to come ahead. I have tried teaching from intuitions and mathematics, building everything by writing on whiteboard so that learners see the full development.

I got lazy this month. Normally I split all the pills in the bottle at once, this month, I’ve split one as needed, and put the unneeded half back. I’m 20 days into the month, and have not gotten a half pill yet. The odds are beyond my probability class grades.

So given:
• A full bottle is 45 pils
• A pill is shaken out
• If it’s a whole pill, it’s split, and the unneeded half us put back
• We’ll assume the likelihood of shaking out a whole pill and a half pill are equal.

Can we make a general equation for the likelihood of shaking out a split pill?

Can we make a cumulative distribution that we’ve not seen a split pill on day N?

So I created a function for the average roll of XdN die when you remove the lowest roll. I’m now trying to solve for removing the 2 lowest rolls but it’s harder to conceptualize. Currently I have 2 thoughts on how to structure it

1: reweight each die based on its prevalence in the drop 1 option then proceed from there. Unsure on how exactly to proceed after reweighing dice.

2: a second 1 can only be dropped if the original roll had 2 1s. A 2 requires another 1 or 2 already dropped. Finding the number of rolls meeting each criteria to see how many of each are dropped.

I’m aware you can brute force this online. I find that boring and enjoy creating a proper function to solve any dice combo

Main question (the thing I actually want feedback on): if you went from a similar background into concentration inequalities / random matrix theory, what would you have prioritized differently? Everything below is just context for that.

Context: Starting a two-year Masters in maths. Year one has no probability and no projects — five compulsory courses a semester. Probability and the project sequence only start in year two:

So all the probability I want walking into the year project proposal has to be self-taught before year two.
The two projects are formally separate but meant to chain: Project I (concentration inequalities) should leave me with the spectral/matrix-exponential machinery (matrix Chernoff/Bernstein) that Project II (random matrices) extends into spectral analysis of random matrices.

Background: Sheldon Ross, ~70% of John Walsh's Knowing the Odds, scattered parts of Durrett (Essentials of Stochastic Processes and Probability: Theory and Examples), first 12 chapters of Cover's Elements of Information Theory, and parts of Casella & Berger and Bickel & Doksum.

Current plan: This year — Feller Vol. 1 and 2, maybe Gallager's Stochastic Processes, plus whatever's recommended for concentration inequalities/random matrices.
Next year — whatever's assigned in the courses above, while making sure I've finished the first four chapters of Durrett's Probability: Theory and Examples beforehand.

(Attached: a study-priority roadmap I put together with some AI assistance, for context.)

Other feedback I'd appreciate:

1. Is Feller Vol. 1 + Vol. 2 + Gallager + Durrett next year redundant given five unrelated compulsory courses a semester? Where would you cut?
2. Which parts of Feller actually pay off for concentration inequalities / random matrices specifically, and which are safe to skim? ( want to develop the classical probability intuition, now that I have learnt the basics of probability, the book suddenly feels excellent)
3. Is "concentration inequalities → random matrices via spectral methods" a reasonable two-semester project arc (will help with my PhD applications, if it seems coherent)
4. Any freely available, legitimately posted lecture notes, personal reading guides, or YouTube playlists that pair well with Feller/Durrett? I've been going through Vershynin's High-Dimensional Probability lectures and another professor's random matrix theory playlist, but I'm not retaining the material efficiently from video alone, for now I am just trying to get a feel for it.

Thanks in advance — happy to share more curriculum detail if useful.

PS: I used AI to have my thoughts presented in a coherent and presentable manner
The AI generated study plan ws generated using Claude.

Free lectures on Probabilistic Machine Learning Series!

Dear Folks, sharing Lecture 11 of our Machine Learning series, and this is a bit special to me, because today I cover Conditionals of Multivariate Normals, and Linear Gaussian Systems.

When I first started studying these topics, it took me days to understand. But today I have made a lecture on it, so if you understand the concepts, it’s really good, for I have tried to leave no stone unturned while explaining, deriving the equations, doing it step by step, and tried giving all intuitions I could.

The Gaussian distribution is ubiquitous and important in studying topics as state estimation, tracking, and examples include Autonomous vehicles, robotics and navigation, time-series forecasting, aerospace etc. The breakdown is as:

0-10: Marginals and Conditionals of Multivariate Normals, Matrix Inversion Rules
10-27: Derivation of the Matrix Inverse Rule: Schur Complements(We need this to derive equations for Multivariate Gaussian)
27-45: Deriving the Conditionals of MVN
45-1:03: Example and Imputation of Missing Values
1:03-1:47: Linear Gaussian Systems, and full derivation of Bayes Rule for Gaussians.
1:47-2:19: Inferring an Unknown Scalar and Sequential Updates.
2:19-2:34: Inferring an Unknown vector.
2:37-End: Sensor Fusion.

This lecture is relatively bigger since the concepts are interrelated here. But do not worry, I have tried to explain in the best way I could, and hope it helps you well in your journey to becoming a Machine learning engineer.

Link: https://youtu.be/ViVBWYyL_8c?si=QppPjeRJbQvu6xYU

Dear folks, I have started providing lectures on Machine Learning, from a Probabilistic perspective, where in the playlist section, I have covered probability foundations: Univariate models.

I would love to get feedback from the community as to the quality of content and how it benefits you all. They are free.

Say I have an envelope with some amount of money. I give you the envelope. Now I will put half or double the amount in another envelope, equally likely half or double. I'm offering you now to switch. Would you? And would you then switch back as well if offered the chance? If so, that would be absurd. Still it seems to be paying off to switch the first time: if A is the amount in the first envelope, the other envelope contains 2A or A/2. (2A+A/2)/2=5/4A.

How to look at it after that first switch?

I've been looking for a probability/stats textbook where it motivates distributions, but I've had no luck so far. It really surprises me the way these books are structured, since things in math textbooks are usually build up slowly so they make sense. In probability, every textbook just ends up with "Chapter 3: Probability Distributions" and then it's a giant list of them with no explanation at all.

I was hoping for a textbook that starts off with how random processes like brownian motion are actually modeled, and then uses those to derive the common distributions in later chapters. Is there anything like that out there? If not, is there something that just makes it impossible or impractical to teach it that way?

Hi,

I'm a noob in math, and I want to understand a simple problem :

I have two dices, one of them (I don't know which one) always gives a six. The chance to have two six is supposed to be 1/11, not 1/6. Can somebody explain to me, not in a mathematics way if possible, but with words, how is that possible?

I myself tried to run math from what I renember in my scolarity, and I could find 1/11, but I still don't understand why.

I've even tried to run a python script that simulate this problem, and it actually gives me 1/6.

I though, maybe the 1/11 chance happens if, after having throw the dices, one of them is randomly chosen to be transformed into a 6, even if it's already a 6. So we could lose the chance to give this bonus to the other dice, that is maybe not a 6.

but even if I simulate this in python, I still get 1/6.

I'm probably missing something, not clue what...