How to Construct Every Randomness | Transform of Random Variables, Random Number Generators

[@jedediahjehoshaphat]

Very insightful, just finished my Advanced Measure Theory paper in university. Wasn't expecting to find applications here, but it surely supplemented my knowledge.

[@Jacob.Cornejo]

wow this makes integration by substitution so clear

[@emanuellandeholm5657]

There is a method in image processing called histogram equalization, which is basically taking an image and processing it such that its histogram becomes more uniform. This can be useful for discarding things like shadows from projections when doing feature detection, as well as a way to salvage overexposed images where the histogram is digitally clipped.

[@neoMushroom]

wow i finally understand what transformation of rv geometrically means because of this video

[@blacklistnr1]

2:38 Really solid into! As a side note: one probability approach I'd like to see more often is setting bounds for classes of the input space. e.g. the ball reaching a point 500m away from the goal? put some bounds on the initial kick energy + wind resistance + time of flight and you get 0% or another class: the ball reaching the top left corner: it needs one of -> which if you diff backwards -> you get of input ranges -> which covers of the input space, so it's now a question of how common/easily do those initial conditions happen This way you get to progressively shape the actual distribution, even if you don't know it (as opposed to usual simplify and "it's just a model") P.S. I just like visual math videos, I don't do math professionally

[@cartatowegs5080]

Guess my sleep is delayed by 26 minutes

[@A_doe_wasting_her_life]

1144 me too 😭

[@tostamista8852]

Are you in my room bc its 4 am and im still gona watch this mf

[@yudoball]

Same

[@karelnarselo]

same here lmao

[@sanjaux]

“Adds to watch later”

[@lees4416]

Little prince distribution. Sounds good, actually

[@fardinahsan2069]

N(100,15), the IQ curve, we meet again

[@Kapomafioso]

My favorite formula of random variable transformation is (from any dimension to any dimension) f_Y (y1, ..., yn) = integral dx1 ... dxm f_X (x1,...xm) delta(f1(x1,...,xm)) delta(f2(x1,...xm)) ... delta(fn(x1,...xm)) where f1, ...fn encode the functional relationship between x1, ..., xm and y1, ...yn. This can go from 1->1 random variable. Or 2->1. Usually n

[@rugbybeef]

[@MsSlash89]

The thumbnail tricked me! As an algebraist, the word “Rng” made me believe there was some Algebraic structure underneath; I cam out disappointed, but also happy to have learned something new!

[@blacklistnr1]

4:42 "times the indicator function from a to b" So this is how mathematicians do ifs :))

[@artmowo2779]

good illustration, thanks!

[@intrepiddt]

Great explanations - thank you!

[@AnythingGoesCodes]

10:02 Does that mean P(-1)=0.5 ?

[@coltonmartin864]

At 14:00 where was the function f_X(x) pulled from. what does this function refer to?

[@APaleDot]

9:00 It should technically be called a "affine" transformation, not "linear"

[@HaramGuys]

All about context. Not everything is in linear algebra language, and in the context of probability theory, linear is more common. Piecewise linear manifolds, linearization of differential equations, all of these concepts technically are affine maps, but no one calls it affine.

[@grayjphys]

I'm wondering if transformations like this could be useful in solving nonlinear odes

[@Ivan_1791]

How are all your videos so great?

[@Vincent-kl9jy]

I would love you to talk about Fokker-Planck Equations in a future video

[@grayjphys]

my thoughts too, especially if things like this could be useful in solving them

[@thenationalist8845]

Very interesting 🤓

[@mehdimabed4125]

Very cool video ! Actually, I'm struggling trying to derive a formula for the CDF (or PDF) of the product of two random variables, and explore some sort of algebra of random variable (I know there is a book with this name but I nothing really satisfying for the product of two random variables....) ; by taking the log maybe ?

[@boium.]

4:10 it's not an integral from -∞ to the dummy variable, but to x. In this case, t is the dummy variable.

[@98danielray]

dummy variable as in the argument of the function. that was pretty understandable.

[@pegrat]

Nice hat in the thumbnail

[@minecrafting_il]

That is just what I wanted!

[@bluekim9771]

1. if we know the function Y =g(X) then we can calculate f_Y(y) from f_X(x) 2. we can generate numbers with algorithm (linear congruential generator) or by natural phenomenon so if the x is generated by phenomeon -> The distribution of x, which is f_X(x) will be made -> but we want the disrtibution be f_Y(y) then we have to find function g where Y=g(X)? is that how we can make a generator for any probability distribution? And why this is realted with inverse integrals?

[@kasiphia]

Interesting.

[@peeepeeepooopooo]

BASED

[@lih3391]

❤ awesome

[@gmdFrame]

You're so cool!!!

[@friggy1899]

Fangraphs sighting!!!!

[@frba9053]

Good to know

[@johncorn7905]

Ok but what is the probability i can get a gf

[@speye]

non-zero 😊

[@montadermajed9456]

Mathematically: 50%

[@johncorn7905]

@@montadermajed9456 what makes you say that

[@johncorn7905]

@@speye i appreciate the confidence

[@montadermajed9456]

@@johncorn7905 i have absolutely no idea

[@DonQuiGoddelaManCHAD]

why did you make the thumbnail a hat

[@HaramGuys]

"I showed the grown ups my masterpiece, and I asked them if my drawing scared them. They answered:'why be scared of a hat?' My drawing was not a picture of a hat. It was a picture of a boa constrictor digesting an elephant." - Antoine de Saint-Exupéry, The Little Prince

[@MouhibBayounes]

Great story especially if you know french. ​@@HaramGuys

[@abdulrhmanaun]

❤❤

[@zhonyss]

I think i'am stupid

[@MatthisDayer]

mersenne twister kinda sucks

[@MatthisDayer]

it's needlessly overcomplicated is not random at all on the lower bits can get stuck producing only zeroes for millions of iterations is hard to seed properly it needs so much memory that it doesn't fit on registers it's kinda slow adds unnecessary binary size in an application using it you really don't need equal distribution in 623 dimensions, 4 is enough for any computation that lasts less than a human lifetime look for xoroshiro128 or xoshiro256 for much better alternatives.