#statstab #501 Prediction intervals for GLMs

Thoughts: Sometimes the prediction of the next dat point can be [0,1]. Not very useful.

#prediction #uncertainty #predictionintervals #glm #binomial #probability

https://fromthebottomoftheheap.net/2017/05/01/glm-prediction-intervals-i/

I'm doing some symmetric monoidal algebra that involves careful counting of some signs of certain permutations. The work depends on a couple of combinatorial identities that I haven't seen anywhere else—do you recognize these (below)?!

The identities involve the "choose two" binomial coefficients.
For ease of typing, I'll use this notation:

[a;2] = binomial(a,2) = a·(a-1)/2
(read "a choose 2")

The two identities are

(I1):
[a+b;2] = [a;2] + [b;2] + ab

and

(I2):
[ab;2] = a[b;2] + b[a;2] + 2[a;2][b;2]

In particular, (I2) means there is a mod 2 congruence
[ab;2] ≡ a[b;2] + b[a;2]
and that's the form that has been particularly useful for me.

Neither of these identities are hard to prove directly from the definition, and they hold for positive *and negative* integers a and b. (That extension to all integers is important for my applications too.)

I've done some internet searching (wikipedia [1,2] and other general references), but I haven't found mention of these particular identities. So, I'm wondering if anyone here recognizes them. (Boosts appreciated!)

Note: These particular binomial coefficients [a;2], for positive a, are also called *triangular numbers*. I'll rewrite (I1) and (I2) in terms of triangular numbers in the next post, in case people will recognize that alternate form (but I doubt it).

[1] https://en.wikipedia.org/wiki/Binomial_coefficient
[2] https://en.wikipedia.org/wiki/Triangular_number

(1/2)

#binomial #triangular #PascalsTriangle

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

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 -