Probability-generating functions • Hacker News Board

Re: Probability-generating functions

#51

Post by Sharlin » Fri, Nov 01, 2024, 8:10 PM UTC

sampo wrote: post #49

Sharlin wrote: post #48

Earlier quoted context omitted.

How so? It’s impossible to express x^n as any linear combination of x^k, k != n.

You have mixed up these two concepts: https://en.wikipedia.org/wiki/Linear_independence https://en.wikipedia.org/wiki/Orthogonal_basis

Okay, I think I get it… I thought one would/could define orthogonality of polynomials in terms of an inner product equivalent to the familiar K^n dot product, ie. sum of pairwise products of equal-degree coefficients, but I guess polynomials just don’t work that way. I can’t say I understand the hows and whys of the "correct" inner product given by Wikipedia :/

Re: Probability-generating functions

#52

Post by sampo » Fri, Nov 01, 2024, 9:27 PM UTC

Sharlin wrote: post #51

sampo wrote: post #49

Earlier quoted context omitted.

You have mixed up these two concepts: https://en.wikipedia.org/wiki/Linear_independence https://en.wikipedia.org/wiki/Orthogonal_basis

Okay, I think I get it… I thought one would/could define orthogonality of polynomials in terms of an inner product equivalent to the familiar K^n dot product, ie. sum of pairwise products of equal-degree coefficients, but I guess polynomials just don’t work that way. I can’t say I understand the hows and whys of the "correct" inner product given by Wikipedia :/

> I thought one would/could define orthogonality of polynomials in terms of an inner product equivalent to the familiar K^n dot product, ie. sum of pairwise products of equal-degree coefficients

Well, you can. But then you essentially get the real coordinate space, and them being polynomials and not just real-valued vectors is just an extra story that you slapped on top, but that has no relevance.

Polynomial vector spaces become useful, when we treat polynomials as functions, and the vector space of polynomials as a subspace of some other space of functions. And with function spaces the inner product is defined as an integral of the product of two functions (possibly with a weight function thrown in).

Or maybe the coefficient-based inner product spaces of polynomials have some uses, too. But they are less common than the function-based (integral) inner product.

Re: Probability-generating functions

#53

Post by bokenator » Sat, Nov 02, 2024, 5:27 AM UTC

schmidtleonard wrote: post #42

creata wrote: post #39

Earlier quoted context omitted.

> the PDF of the sum of two random variables is the convolution of the original PDFs (Probably obvious to everyone reading, but the variables should be independent.)

But I'd rather assume the variables are independent and then blame statistics when I get the wrong answer!

This is a good place to use cumulants. Instead of working with joint characteristic functions, which gets messy, it lets you isolate the effects of correlation into a separate term. The only limitation is that this doesn't work if the moment doesn't exist.

Re: Probability-generating functions

#54

Post by Sharlin » Sat, Nov 02, 2024, 10:45 PM UTC

sampo wrote: post #52

Sharlin wrote: post #51

Earlier quoted context omitted.

Okay, I think I get it… I thought one would/could define orthogonality of polynomials in terms of an inner product equivalent to the familiar K^n dot product, ie. sum of pairwise products of equal-degree coefficients, but I guess polynomials just don’t work that way. I can’t say I understand the hows and whys of the "correct" inner product given by Wikipedia :/

> I thought one would/could define orthogonality of polynomials in terms of an inner product equivalent to the familiar K^n dot product, ie. sum of pairwise products of equal-degree coefficients Well, you can. But then you essentially get the real coordinate space, and them being polynomials and not just real-valued vectors is just an extra story that you slapped on top, but that has no relevance. Polynomial vector s…

Yeah, that's what I figured. Thanks for your comments, I learned something new!