Thanks for the video. I am unsure as to why you motivate the expected value as a linear function that must be equal to the probability of the event for the case of an indicator. Why is that more fundamental or why not just observe that from the definition?
You are welcome :) From the perspective of introducing new theory, it is nice if you can motivate it, i.e., argue why the definitions are natural. That is why I took this axiomatic approach: I wanted to make the argument that this is really the only sensible definition of expectation. Since both characterizations of expectations are equivalent, they are equally fundamental and doing it this way was just a personal choice.
I don't think that that is totally evident. In the discrete case, I would even agree that it is not more evident than the definition of expectation. However, it is a nice way to think about expectations and in hindsight I probably presented it with that order since we can use the ideas to define (lebesgue) integrals which we need in order to define the expectation of a general (not necessarily discrete) random variable
@@calimath6701 it would be awesome if you made a video connecting that to general continuos variables, and the analogous way of defining X as a sum of indicators in that case
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I don’t know how I discovered this channel but I am so glad I did 🔥
Glad you enjoy the videos. That means a lot to us
This is really great, better than A LOT of measure theoretic so called "Introduction" textbook! Please make more :)
Great video. A Second video on this topic would be highly appreciated.
Funny intro. These type of videos are amazing. More of then would be amazing.
Thank you so much 🙌🏾
Thank You
Thanks for the video. I am unsure as to why you motivate the expected value as a linear function that must be equal to the probability of the event for the case of an indicator. Why is that more fundamental or why not just observe that from the definition?
You are welcome :)
From the perspective of introducing new theory, it is nice if you can motivate it, i.e., argue why the definitions are natural. That is why I took this axiomatic approach: I wanted to make the argument that this is really the only sensible definition of expectation.
Since both characterizations of expectations are equivalent, they are equally fundamental and doing it this way was just a personal choice.
@@calimath6701 But wanted to ask you why is it something evident that E(I{A})=P(A) must hold?
I don't think that that is totally evident. In the discrete case, I would even agree that it is not more evident than the definition of expectation. However, it is a nice way to think about expectations and in hindsight I probably presented it with that order since we can use the ideas to define (lebesgue) integrals which we need in order to define the expectation of a general (not necessarily discrete) random variable
@@calimath6701 it would be awesome if you made a video connecting that to general continuos variables, and the analogous way of defining X as a sum of indicators in that case