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Why the "central limit" is a normal distribution
🕑 Added 2023-07-10 18:49:28 +0000 UTCHere's an early view for the next video, capping off the mini-series about normal distributions. I have just one to-do stub left in there for myself, but let me know if you catch anything else that should change before the final release.
Comments
Daksha Vaid-Kwinter
Differential entropy is just the continuous analogue of discrete entropy.
Stuart Jones
Hey Grant! I think it'd be super interesting to see a video/series on discrete calculus (so many mathematics majors never get to see the intricacy and beauty of it!) and use the connections with continuous calculus to give intuition for the binomial approximation of the normal distribution. Just some food for thought for a different angle to approach!
Craig Falls
I had been meaning to try to create a proof of CLT based on the intuition that convolution increases entropy "faster" than variance, and Gaussians maximize entropy for a given variance. Partly I was frustrated that most justifications of the Gaussian distribution go to two dimensions, and it seems like that shouldn't be necessary. Interesting to hear that this is a standard argument, although apparently it uses something called "differential entropy" and I don't know what that is.
Thomas Schneider
Correction: https://www.youtube.com/watch?v=d_qvLDhkg00 9 minutes 3 seconds https://www.youtube.com/watch?v=d_qvLDhkg00&t=9m3s right side second word: transformatoin -> transformation
I must admit that I am somewhat underwhelmed. Showing that the Gaussian function is a fixed point of repeated convolution seems like a first, small step. I do not feel any closer to understanding why the CLT must be true for (almost) arbitrary distributions. I was hoping to see some intuition for why there is a contraction towards this this fixed point. Then the CLT would follow directly via the Banach fixed-point theorem. Maybe the contraction could be shown in the Fourier domain, since convolution turns into multiplication. And the Fourier transform of a Gaussian function is another Gaussian function.
