20. Definitions and Inequalities

[@timdelise6326]

How many MIT courses are there with no videos, and Prof. Strang comes through with an incredibly relevant course with full course videos. Amazing.

[@georgesadler7830]

Professor Strang thank you for explaining and analyzing Definitions and Inequalities from probability and statistics. The pictures and examples really help me to understand the quality of this lecture.

[@ziqichen4654]

For chebyshev, on the left hand side, the right bracket should close after ">=a".

[@allyourcode]

Also, at the end, he was supposed to say that P[Y >= a^2] instead of E[Y]

[@1290DR]

Mr. Professor - you are great!

[@ankits2513]

You sir are an Inspiration:)

[@cyberneticqualanaut7207]

I can see and prove a covariance matrix with x and y independent is a positive definite matrix. But when x and y are dependent, I don't think the covariance matrix V is positive semidefinite. In the 2x2 case uT*V*u could be less than zero for the case when u = [1; -1] for instance.

[@christianjimenez1877]

Brilliant!!!

[@tmusic99]

I am missing a reference to the population when explaining expected mean and expected variance. Then you sample the population for a sample mean and/or a sample variance.

[@rusty-b3p]

11:55 when you try to guess answer on multiply choice test

[@quanyingliu7168]

Proof of Markov Inequality: en.wikipedia.org/wiki/Markov%27s_inequality#Proof_in_the_language_of_probability_theory

[@1u2t3]

The example for Markov inequality seems not proper. For sequence 1,2,3,4,5, the mean can be 1 only if P2, P3, P4, P5 are all 0

[@seamus9898]

I agree it's a weird simplified example but the math still checks out

[@strahilpastuhov1886]

P1 need not be 1, so P2 etc. can be above zero and the mean still be 1.

[@PrzemyslawSliwinski]

@@strahilpastuhov1886 Nope. In this example you have that 2p₂ + 3p₃ + 4p₄ + 5p₅ = 1 - p₁ = p₂ + p₃ + p₄ + p₅. That is, you get p₂ + 2p₃ + 3p₄ + 4p₅ = 0.

[@PrzemyslawSliwinski]

​@@seamus9898 Yup, it is almost like a dice example.

[@linelendur]

Yeah, I paused the video just after he collected terms and tried to move forward myself (by collecting (p1+...+p5) together) and immediately hit this issue. That said, if you replace "1" with E[x] in both example the result he's showing still falls out, as "E[x]/3" instead of "1/3"

[@brendawilliams8062]

Thx.

[@ranjit8721]

In 18.05, the camera movement was very nice. In 18.065, I think the camera person is not that professional.

[@sicongliu1484]

What is 18.05? Could you please share the links and I am very interested. Thanks!

[@deepaks.m.6709]

Another useful resource: metacademy.org/graphs/concepts/chernoff_bounds#focus=chernoff_bounds&mode=learn