3 ms·

That doesn't sound right. If P(X) is the vector {0.5,0,0.5} and P(Y) is {0.5,0.5,0}, P(X)P(Y) is {0.25,0,0} and that's both not normalized and clearly not the d

That doesn't sound right. If P(X) is the vector {0.5,0,0.5} and P(Y) is {0.5,0.5,0}, P(X)P(Y) is {0.25,0,0} and that's both not normalized and clearly not the distribution for max(X,Y). Did you get that from an LLM?

srean 1y ago
You are using PMFs. I meant and wrote distribution function aka cumulative distribution function. They are closed under products. > Did you get it from LLM LOL. There must be a fun and guilty story lurking inside the accusation. On a more serious note, I would love it if LLMs could do such simplifications and estimations on their own.
- deleted 1y ago
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- whatshisface 1y ago
  Distributions can be either PDFs or CDFs. To be honest I'd never heard of assuming that a distribution was a CDF unless otherwise specified.
  - srean 1y ago
    May I raise you a https://en.m.wikipedia.org/wiki/Distribution_function https://en.m.wikipedia.org/wiki/Distribution_function It's right in the title. In probability theory, integration theory, as well as electrical engineering, "distribution function", unless further clarified, means that cumulative thing. In math, nomenclature overloading can be a problem. So context matters. In the context of dirac delta, distribution means something else entirely -- generalized functions. Oh! So sad you deleted your point about densities. One can only laugh at and enjoy these idiosyncrasies of nomenclature. In Electrical Engineering one uses j for imaginary numbers because i is taken (by current).
    - deleted 1y ago
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  - math_dandy 1y ago
    This is a natural point of confusion. The true (IMO) primitive concept here is the probability measure. Probability measures on the real line are in canonical bijection with CDFs, the latter being axiomatizable as càdlàg functions (see https://en.wikipedia.org/wiki/Càdlàg https://en.wikipedia.org/wiki/Càdlàg) asymptotic to 0 (resp. 1) at minus infinity (resp. infinity). On the other hand, not every probability measure has a density function. (If you want the formalism of densities to capture all probability measures, you need to admit more exotic generalized functions à la Dirac.)

srean 1y ago

You are using PMFs. I meant and wrote distribution function aka cumulative distribution function. They are closed under products. > Did you get it from LLM LOL. There must be a fun and guilty story lurking inside the accusation. On a more serious note, I would love it if LLMs could do such simplifications and estimations on their own.

deleted 1y ago
[deleted]
whatshisface 1y ago
Distributions can be either PDFs or CDFs. To be honest I'd never heard of assuming that a distribution was a CDF unless otherwise specified.
- srean 1y ago
  May I raise you a https://en.m.wikipedia.org/wiki/Distribution_function https://en.m.wikipedia.org/wiki/Distribution_function It's right in the title. In probability theory, integration theory, as well as electrical engineering, "distribution function", unless further clarified, means that cumulative thing. In math, nomenclature overloading can be a problem. So context matters. In the context of dirac delta, distribution means something else entirely -- generalized functions. Oh! So sad you deleted your point about densities. One can only laugh at and enjoy these idiosyncrasies of nomenclature. In Electrical Engineering one uses j for imaginary numbers because i is taken (by current).
  - deleted 1y ago
    [deleted]
- math_dandy 1y ago
  This is a natural point of confusion. The true (IMO) primitive concept here is the probability measure. Probability measures on the real line are in canonical bijection with CDFs, the latter being axiomatizable as càdlàg functions (see https://en.wikipedia.org/wiki/Càdlàg https://en.wikipedia.org/wiki/Càdlàg) asymptotic to 0 (resp. 1) at minus infinity (resp. infinity). On the other hand, not every probability measure has a density function. (If you want the formalism of densities to capture all probability measures, you need to admit more exotic generalized functions à la Dirac.)