What is the Central Limit Theorem?

Gives a brief explanation of the Central Limit Theorem, and shows an example where the individual random variables have uniform distributions.
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[@AlphaSquarehunt]

Brilliant explanation Professor! Intuition is what teachers lack while explaining stats , you are one of the few gems capable and smart enough to integrate it. Thank you!

[@iain_explains]

I'm so glad it was helpful!

[@fifaham]

Very few who can explain those concepts in a simple and understandable fashion the way you do. Thank you very much for your time and efforts.

[@iain_explains]

Glad it was helpful!

[@manikandanshanmugam7480]

Very good explanation

[@iain_explains]

Thanks for liking

[@rabbulkhan7132]

Could anyone kindly explain what does "under weak conditions" mean?

[@charlesgeorge85]

Great video...

[@fifaham]

Man... how lucky are those who studied communications systems under your supervision.

[@abdullahsy7072]

Great video

[@tilak1431]

Thanks for the video. Just a point about the same mean condition, did you mean the overall mean of the RVs? Because in your example the means are changing with additional RVs when you add them (20, 30...) and then you subtract them.

[@iain_explains]

It refers to the means of each individual random variable that is added in the sum (ie. X_1, X_2, X_3, ... in the example I showed). As I mentioned at the 4:43 mark, I only showed the direct convolutions, without subtracting the means, because that was easier to show in the drawing. But it doesn't matter if you subtract the means as you go in the sum, or do it all at the end. It's the same either way.

[@yasserothman4023]

Thanks for sharing, what if the random variables were subtracted instead of added ? What are those conditions for this therom to be valid and what should make them strong instead of weak ?

[@iain_explains]

Subtracted from what? If you mean they are all negative, then it's exactly the same, but where the random variables have their probability mass on the left hand side (ie. the negative values). In other words, you are adding negative numbers. If, on the other hand, you mean that the first RV in the sum is positive, and then all the others are subtracted from it, then that's not covered by the CLT, but I'm not sure there are many practical examples of this to make it relevant. In terms of your question about the conditions. We actually prefer to have "weak" conditions, rather than "strong" conditions. Weak conditions are not very restrictive. Strong conditions are very limiting.

[@anaya1012]

Thank you

[@iain_explains]

You're welcome.

[@hasanbayazit1689]

informative

[@lupgavgavu]

Hello dear Professor, I couldn't understand why convolution is used. What should i investigate for this ? Thank you for all.

[@iain_explains]

If you watch from the 8:40min mark of the following video, it explains that when you add random variables (which is what is being done in the Central Limit Theorem), then the resultant pdf is found from the convolution of the pdf's of the component random variables: "What is Convolution? And Two Examples where it arises" kzbin.info/www/bejne/jmPGe2usdshjg7c

[@lupgavgavu]

@@iain_explains Thank you very much, Dear Professor. İt was be very helpfull. :)

[@hemantsharma2416]

Can RVs X1, X2....Xn have different distributions or not?

[@iain_explains]

For the standard central limit theorem, yes, all the RVs need to come from the same distribution. But there are extensions/variants of the theorem that give certain results about the limiting mean etc, for some cases on non-identical distributions.

[@hemantsharma2416]

@@iain_explains Thank You....