/sci/ - Science & Math

https://youtu.be/qr1Odx1iZGs
Tell me if I should be shorter (less elaborate) with those.
There are at least 2 exercises at end of Chapter 2 that can be approached with having read the first two sections of Chapter 2.

Some definitions are made (not too much detail, though), and some equalities are to be shown.

Conditional probability:
[math] p(x \, | \, y) := \dfrac { 1 } { p(y) } p(x,y) [/math]

Let
[math] p(x \, | \, y,z) := p(x \, | \, z) \, p(y \, | \, z) [/math]

and prove that this is equivalent to
[math] p(x \, | \, z) := p(x \, | \, z, y) [/math]
[math] p(y \, | \, z) := p(y \, | \, z, x) [/math]

and after this in the text, there follow some probability laws that are to be proven as well.

The other exercise is about
[math] Cov[X] := E[X-E[X]]^2 [/math]

One should show that
[math] Cov[X] = E[X^2] - E[X]^2 [/math]