My favourite application of "random variables as vectors" is that Cauchy-Schwartz with Cov as the inner product allows us to interpret the correlation coefficient as the cosine of the angle between two random variables
I did an introductory stats course this year (alongside a more advanced math course) and when I noticed this, I had one of my favourite eureka moments in recent memory. Definitely makes stats a lot more interesting to think about
I'm in no way saying this in a smug way, but all of that is just a consequence of the fact that any function can be seen a vector(yes, even continuous ones). No wonder things like "orthogonal functions" exist even though there is no such thing as an angle between functions. It's all fucking connected and I love it so much that's furious. Btw, I love the way you came across it and your presentation style
Thanks for the comment! Yeah, as I mentioned in the description, it's quite related to the idea of constructing functions via simple functions in measure theory. The important part here is to find what the basis is and in my case I sorta stumbled across the basis which also turns out to be quite nice. And yes! I love the interconnections and it's crazy we can define geometric notions beyond geometry. I kept it real analysis free because I thought it would muddle up the story I wanted to tell. Thanks again!
My friend and a comment mentioned that it's related to the correlation coefficient? I thought the X_a's were the projections! I'd love to know the interpretation in general
My favourite application of "random variables as vectors" is that Cauchy-Schwartz with Cov as the inner product allows us to interpret the correlation coefficient as the cosine of the angle between two random variables
I did an introductory stats course this year (alongside a more advanced math course) and when I noticed this, I had one of my favourite eureka moments in recent memory. Definitely makes stats a lot more interesting to think about
I'm in no way saying this in a smug way, but all of that is just a consequence of the fact that any function can be seen a vector(yes, even continuous ones). No wonder things like "orthogonal functions" exist even though there is no such thing as an angle between functions. It's all fucking connected and I love it so much that's furious.
Btw, I love the way you came across it and your presentation style
Thanks for the comment! Yeah, as I mentioned in the description, it's quite related to the idea of constructing functions via simple functions in measure theory. The important part here is to find what the basis is and in my case I sorta stumbled across the basis which also turns out to be quite nice. And yes! I love the interconnections and it's crazy we can define geometric notions beyond geometry. I kept it real analysis free because I thought it would muddle up the story I wanted to tell. Thanks again!
This is like a 1TV of shock to my brain. Love it
Really cool!
Was just studying Dirichlet distribution
Thank You
Nice video! Just wait until you find out what vector projection means with this inner product....
My friend and a comment mentioned that it's related to the correlation coefficient? I thought the X_a's were the projections! I'd love to know the interpretation in general
Would you care to explain?
@@Sophia_Howell Projection is conditional expectation :) so like vec v projected onto vector u is the same as E[V | U]
Oh that's extremely cool! That makes sense, because you're asking if u happens then how likely is v, which is maximized in the direction
@@MihaiNicaMathty!
this is some matrix level shit.
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that was really cool
i love thisss
this is great! please keep making more!!
Thank you :)
Wa-pa-pa-pa-pa-pa-pow!