Using probability to find the volume of hyperspheres (and other weird balls)

[@johnchessant3012]

here from Mathemaniac! this is really awesome

[@viswapriyamisra]

WOW!! What an explanation...loved it!

[@serious-goober]

Akiva is truly one of the goats

[@RUDRARAKESHKUMARGOHIL]

Also why i am not able to get the same figure in geogebra as you get for Lp balls...? Also if possible can you explain the processes from 13:28 to 14:28 because i feel i am completely lost here... as later you took a [-1,1]^3n as cubes and i am left with why and what and how ....

[@mathematimpa]

Roughly, the idea is that the sum of 2 squares was uniform inside a circle, because the scaling law made everything nice in 2d. The way things click in 3d is if you use the sum of 3 cubes (and in general in nd, you need the sum of n n-th powers to get a uniform)

[@RUDRARAKESHKUMARGOHIL]

Sorry but I didn't get it...😢

[@NehmuloseDoomish]

Muito brabo, curti demais a ideia. Acabei de ver que o canal é brasileiro, chega ser irônico um brasileiro ensinando pro outro em inglês kkkkk. Parabéns pelo vídeo!

[@mathematimpa]

Valeu, eu tenho planos pra fazer uma versão em português do canal, mas ainda vai levar um tempo 😁

[@RUDRARAKESHKUMARGOHIL]

At 14.08 you said "sum of 3 cubes is uniform in interval 0 to 1" so you meant that probability sum of 3 cubes is directly proportional to there volume as you said random points/uniform is that if we pick a point in a dimension say 1D then the the probablity of it on the interval say a to b is proportional to length of ab carrying the analogy here you are trying to say that probability of point in 3D in p-ball is proportional to volume of p-ball. BUT what is the value of volume this weird shaped p balls we dont have any formula for them so how we came accross c ? I am some what confuse here so can younplease sort it out ?

[@mathematimpa]

The idea is that if you draw 3 numbers x,y,z in [-1,1] AND these numbers are such that |x|^3 +|y|^3 + |z|^ 3 < 1, THEN |x|^3 +|y|^3 + |z|^ 3 has the cumulative of a number drawn in [0,1]. Being inside the unit Lp-ball is important for the result to be uniform (technically, we'd need to use conditional probability to clean everything, but I thought it made things way too long)

[@RUDRARAKESHKUMARGOHIL]

But why is it necessary to be inside Lp ball as if it is any regular equation like x^2+|y|+|z|^3

[@1.4142]

17:39 typo with the dx

[@andretimpa]

oops

[@RUDRARAKESHKUMARGOHIL]

Wait a minute you made an equivalent of 4D hypercube as 2 , 2D squares but you did not created an equivalence yet i.e not shown that this is possible and right as you have shown in F A( C) and F x^2+y^2 (C) i.e A and x^2 + y^2 are equivalence....so how is points in a hypercube is same as points drawn independently in 2 squares ? Am i missing at something...if so...sorry.

[@mathematimpa]

The idea is that you are drawing 4 numbers at random. You can interpret these 4 numbers as being a point in 4d or two pairs of points in 2d. Being 100% technical you need to assume the numbers are being drawn independently too, but I thought this would just add an extra concept that is already kinda intuitive.

[@RUDRARAKESHKUMARGOHIL]

Ty ❤

[@RUDRARAKESHKUMARGOHIL]

How is volume of hypercube 16 ? you said by multiplying all sides but all sides are 1...so it should be 1. Sorry if the doubt is obvious but currentl i am not getting it ...

[@mathematimpa]

@@RUDRARAKESHKUMARGOHIL The sides go from -1 to 1, so each side has lenght 2

[@RUDRARAKESHKUMARGOHIL]

Thankyou...also as you sated Two variables(random) with same cumulative distribution are equivalent...I wanted to know that why is it so...and what is the equivalence here ? thankyou also can we do it for area of sphere I.e disintegrating it in 1d line and a 2d square ? Also great work best wishes for your SOME WIN 🏆 👍 .

[@mathematimpa]

If X and Y have the same cumulative, this implies a few things. First, the probabilities that X=a and Y=a is the same for every a. Also the probability that X is in [a,b] and that Y is in [a,b] are also the same. This means that you pretty much can't tell them apart from their values. Every property will be the same and that's why you can use them interchangeably.

[@RUDRARAKESHKUMARGOHIL]

Also could you provide a intuition for volume of Hyperpyramids...i.e about the division by n in formula i got the intuition about the base through the video. ...Sorry for so much doubts.😅.

[@mathematimpa]

@@RUDRARAKESHKUMARGOHIL The division by n thing requires calculus to fully explain. The idea is that you can consider a section of the pyramid parallel to the base, you get a scaled down version of the base. You can use the scaling law to write this volume as a function of the base volume. If you integrate this section volume you get the volume of the whole thing (The 1/n ends up coming from the scaling law, but you need to put it down in paper to see this)