STATISTICS 1 (2002-2010)
C1 Representation of Data
C2 Permutations and Combinations
C3 Probability
C4 Discrete Random Variables
C5 The Normal Distribution
the central limit theorem is a frankly magical theorem. i'll assume everyone reading this post is familiar with it, i'm speedrunning out my thoughts, i don't want to spend too much time on this. but basically yeah why should everything add up to the normal distribution? why should the mean of basically everything be normal?
so this bothered me for a long time, it was always kinda insane. i learned about it, and then a while later i learned how to prove it, but i still didn't get it. but then i watched some 3B1B videos (all kudos to them btw) and then specifically i saw this one comment on a video and it all clicked.
basically: what's so special about the normal distribution? like, why *that* one in particular?
well, there's this other property of normal distributions, which may seem unrelated at first but is actually the key to the whole thing: if you add two normal distributions, you get another normal distribution.
and this is the key to the whole thing!
because it means that, when you're doing sums of random variables, normal variables are a *fixed point*.
so at least for me this instantly finally made everything click - things converging to fixed points is just so incredibly natural, it comes up everywhere.
to go into a little more detail, distributions where if you add two of them together you get a third one of the *same type* are called stable distributions, and it turns out that the normal distribution is special among stable distributions because it's the unique one with finite mean and variance - all the others, Cauchy, Levy, etc. don't satisfy this condition. i don't understand why but yeah.
so then it makes intuitive sense that - if you keep averaging lots and lots and lots of variables which have finite means and variances together and converge to anything at all - it *must* be to the only fixed point which has finite mean and variance:
the normal distribution.
I'm gonna be thinking about Ghost forcing his way into your life because he can't date people normally for a while. The man cannot date, he can't. He doesn't have the right settings, you're either his or you're not and if he decides you're his then why should he mess around with all that dating nonsense. Just skip straight to being partners, straight to moving in together, you wanna go down to the courthouse? He's already filled out the forms you can get married today. No? That's fine, maybe tomorrow.
An asteroid wipes out 75% of all species on earth.
There are 1000 species. The population of each is normally distributed around 1 million with SD of 100,000.
A species is extinct if reduced to 0 individuals.
What is the maximum and minimum percent reduction in the total population of individual creatures on earth?
If every species is equally likely to end up extinct (that is, going extinct is not dependent on population size) what is the expected percent reduction in the total population of individual creatures on earth?
My favorite thing about Dungeon Meshi is the way it repeatedly establishes how eating monsters is relatively normal, despite everyone always losing their minds over it. Oh man, Laios wants to get through the dungeon by eating monsters?? How horrib- oh here's this dwarf that's been doing that for 10+ years. Eurgh, gem bugs?? That's so weird- oh if you make it a jam it's considered a delicacy in several parts of the world. Mermen?? No way, that's way too similar to mermaids to ever- oh would you look at that, mermaids eat them. Huh