ohohoho 🔮

#binomial #72

https://sowe.li/binomial/072_/

on orbs and orblikes

One day, one decomposition
A138389: Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1)

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/Binomial_primes.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Binomial_primes.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #binomial #primes #PrimeNumbers #coprime #graph #threejs #webGL

i'm up to 67 posts of binomial, my blog/newsletter. the focus has drifted since i started it, but it has remained a place for reflections and play.

someone i met 10+ years ago sent me a message saying they read and enjoyed the stuff so far. perhaps you will, too!

#writing #binomial #shortfiction

https://sowe.li/binomial/

One day, one decomposition
A121943: Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A121943.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A121943.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #central #binomial #coefficient #divisible #graph #threejs #webGL

Try to prove the following two results that relate the harmonic numbers to the golden ratio. Have an excellent weekend.

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^n}=2\sqrt5\ln\varphi\]

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^nn}=\frac{2\pi^2}{15}-2\ln^2\varphi\]

where \(\varphi=\frac{1+\sqrt5}2\) is the golden ratio; and \(H_n=\left(1+\frac12+\frac13+\ldots+\frac1n\right)\) is the \(n\)-th harmonic number.

#GoldenRatio #HarmonicNumbers #HarmonicNumber #Logarithm #Pi #Summation #Math #Sum #InfiniteSum #Binomial #BinomialCoefficient #Maths #WeekendChallenge

In the process I also thought of a #binomial #coefficient interpretation for choosing k things out of a bag of n objects when the objects can be put back and picked again. In probability problems, this is referred to as picking "with replacement".

Usually, when we say, n choose k, we do not allow repeated choices, every chosen object has to be chosen once. In this case, I want to allow replacements but at the same time, I want to keep using my trusted n choose k idea.

Here's my way around this: We'll still be picking k things, but we'll pick them not from 1 to n but from the set {1, 2, ..., 𝑛, 𝑟₁, 𝑟₂, ..., 𝑟ₖ₋₁}. That is, in addition to the n objects, we add what I'm calling "replacement tokens" 𝑟ₓ. If any of the 𝑟ₓ gets picked, then it is interpreted as the 𝑥th choice was put back and you are now choosing to pick that again. Since the 𝑘th choice is not put back, we only need replacement tokens for choices 1 to k-1.

With these replacement tokens, the problem becomes a standard choose k things out of this set, which we can resolve using the binomial coefficient to get: \({ n+k-1 \choose k }\).

I believe the standard approach to this is via #StarsAndBars but I liked the idea of this "replacement token". Admittedly, I didn't want to use stars and bars here and made up some stuff which I happened to like. :)

#math #counting #proof

Write the n-th triangular number as n^Δ
Find that while (a+b)^2 = a^2 + 2ab + b^2,
(a+b)^Δ = a^Δ + ab + b^Δ
#math #maths #binomial

O tabuleiro de #Galton : 3.000 bolas de aço caem por 12 caminhos ramificados e se combinam em uma distribuição de curva em sino. Cada bola tem 50% de chance de seguir cada caminho, seguindo a distribuição #Binomial -

twitter.com/Rainmaker1973/stat… -
RT Massimo - #Normal -
[ Já esteve aqui. É tão bom que voltou ] - 2020 -

One day, one decomposition
A034856: a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A034856.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A034856.html

#decompwlj #maths #mathematics #sequence #OEIS #javascript #php #3D #numbers #binomial #graph #threejs #webGL

New LabPlot tutorial:

https://youtu.be/g0OQrAVwsTw

In this short video you'll learn how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #ProbabilityDistribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood #SciDAViS

New LabPlot tutorial

https://tube.kockatoo.org/w/9Kmqeefqw35EpEZj914N5p

In this short video you'll learn how to how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #Probability Distribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood

@kde@lemmy.kde.social

Do you want to know how to fit a probability distribution to data? Watch the latest @LabPlot video tutorial.

https://www.youtube.com/watch?v=g0OQrAVwsTw

#DistributionFitting #ProbabilityDistribution #Distribution #StatisticalDistribution #Gaussian #Log-normal #Probability #Poisson #Binomial #Exponential #MaximumLikelihood #LabPlot

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

Le fun est total.

(c'est un vrai jeu)
(enfin c'est un logiciel, que j'ai acheté avec mon argent, dans le but de me détendre, sur steam)

#infiniteturtles #zachtronics #zachlike #binomial

I recently read a very interesting paper on Leakage-Abuse Attacks against Order-Preserving Encryption (OPE) schemes and Order-Revealing Encryption (ORE) Schemes.

In this paper, the researchers show how the widely used encryption schemes are inadequate. Here are some snippets from the paper.

Order-preserving encryption (#OPE) - ensures that Ek(m1)<Ek(m2) for m1<m2 and Ek the encryption algorithm. Most widely used scheme is #BCLO.

Order-revealing encryption (#ORE) - reveals ordering relations by way of a public comparison function that operates on pairs of plaintexts. Most widely used scheme is #CLWW.

Popular belief is that OPE and ORE schemes remain secure in practice for plaintext data drawn from larger domains, and practitioners could simply avoid using OPE for small-domain data.

The researchers used a non-crossing attack (min-weight non-crossing #bipartite matching) which runs in only a few hours, even for the largest target dataset, against real-world datasets using the BCLO scheme to encrypt a set of first names.

Using this attack they were able to recover almost half the data set. The leakage was even worse for last names, with almost 97% of last names trivially recoverable.

#Composition of the two (BCLO & CLWW) schemes does #decrease attack accuracy but is still far from providing acceptable security.

Exploiting known plaintexts is even easier.

Attacking frequency-hiding schemes - #Kerschbaum recently introduced a scheme that hides frequency information. However, a “#binomial” attack performs reasonably well, recovering on average 30% of first names and 7% of last names. Notably, it recovers majority of high-frequency plaintexts (despite not having frequency information leaked), suggesting these plaintexts are particularly poorly protected by any order-revealing scheme.

In terms of countermeasures, an obvious suggestion is to move towards less leaky schemes, such as those that only reveal order, including Kerschbaum's scheme and the more recent #Boneh et al. scheme based on #multilinear maps. Unfortunately in most settings there exists inherent #challenges to deployment of these schemes. Kerschbaum's scheme is relatively efficient, but requires client-side state which impedes scaling. The Boneh et al. scheme has ciphertexts larger by 10 orders of magnitude than BCLO ciphertexts and requires tens of minutes to compute encryptions.

#encryption #quantumcomputing #quantumcryptography

Bayesian : using a proper prior, e.g., #Poisson or Negative #Binomial distribution, where closed formula for posterior mean and posterior variance can be obtained.
German tank problem

Bayesian : using a proper prior, e.g., #Poisson or Negative #Binomial distribution, where closed formula for posterior mean and posterior variance can be obtained.
German tank problem

#standarddeviation or luck factor of a simple game like #Roulette can be simply calculated because of the #binomial distribution of successes sqrt(npq)
While house edge of skill game like Spanish 21 is calculated assuming optimal choices

#standarddeviation or luck factor of a simple game like #Roulette can be simply calculated because of the #binomial distribution of successes sqrt(npq)
While house edge of skill game like Spanish 21 is calculated assuming optimal choices

Coronavirus-Bekämpfung: Wie Sars-CoV-2 in Deutschland aussterben kann - Spektrum der Wissenschaft
https://www.spektrum.de/news/wie-sars-cov-2-in-deutschland-aussterben-kann/1741310?utm_source=pocket-newtab-global-de-DE

#Coronavirus #covid-19 #Sars-coV-2 #bekämpfung #superspreader #statistik #stochastisch #aussterben #stochastic_extinction #binomial #poisson #neuseeland #social_distancing #Medizin