Properties of leverage points in regression with proofs - note typo

The properties of leverage points in regression are:
sum of leverages over observations equals the number of parameters in the model (intercept plus number of slope parameters)
TYPO: Var(residual) = sigma^2(1-h_ii) not sigma^2sqrt(1-h_ii)
each leverage point takes a value between 0 to 1 inclusive.
I present the proofs. Along the way I show you that positive semi definite matrix has non negative diagonal elements.

[@xiefeng7240]

thanks! This is so much better than the tutor Phil who teaches me ST300!

[@williambao9164]

absolute banger

[@jessxu4153]

Shouldn't the variance of residuals be equal to sigma ^2 * (1-hi) ?

[@PhilChanstats]

Yes, thanks. At 1:17 the variance of the residual of the ith point is sigma^2*(1-h_ii). Basically, I should not have a square root on the screen over 1-h_ii
ALSO
at 11:12 on the 4th line, the equal sign should be pointing at e'e as it's e'e=1

[@micahdelaurentis6551]

this is great, thank you!

[@Delahunta]

Hi your I liked your video. I had a question though. My homework states if n=p show that H is the PxP identity matrix. What is his and Yi_hat. Our teacher never taught us this and I was wondering if you could show this?

[@PhilChanstats]

If n=p then X is square and if it's invertible then what you say is true - that the hat matrix is the identity matrix. This is just a curiosity, not a result of practical interest as mostly we have n > p in courses in basic regression, (or p