In this video, we aim to construct random number generators of any probability distribution. We introduce the theory of random variables and transform of random variables in order to construct random number generators of any distribution.
In this video, the linear congruential generator is introduced.
https://en.wikipedia.org/wiki/Linear_congruential_generator
However, the much more popular method used today is the Mersenne Twister, which is significantly harder to understand, and it is beyond the scope of this video.
https://en.wikipedia.org/wiki/Mersenne_Twister
Two main techniques used in this video are the probability integral transform and the inverse transform sampling.
https://en.wikipedia.org/wiki/Probability_integral_transform
https://en.wikipedia.org/wiki/Inverse_transform_sampling
My other video on a more rigorous definition of random variables:
https://youtu.be/-X9u9oYvGYk
Chapters:
00:00 Intro
02:38 Continuous Distributions
05:19 General Theory of Random Variables
07:47 Piecewise Linear Transformation of Random Variables
10:10 Nonlinear Transformation of Random Variables
13:06 Geometry of Transformation of Random Variables
16:24 Linear Congruential Generator
19:54 Probability Integral Transform and Inverse Transform Sampling
23:30 Generalized Inverse
25:06 Transform Between Any Two Distributions
26:13 Outro
Music🎵:
Titanium - Mittsies https://youtu.be/K_7K7v2KGYU
Pocket's Lookin' Light - Gareth Coker · Riot Forge https://youtu.be/kifzZ0Txkfs
Runnin' From Lariette - Gareth Coker · Riot Forge https://youtu.be/LEckRjROY00
Machina del Diablo - Stephen Rippy
Driftveil City Funk Cover - Sean R. Hanson https://youtu.be/Gax8DLUzWYo
Corrections:
19:45 Mersenne Twister is not the best RNG today
22:56 CDF not PDF