Markov inequality explain??

[somber urchin]

This is my professors proof. What is this even about, where do you even use this, how do you interpret this and can someone explain the proof?? I don't get where does E[X 1lambda>=0] >= E[lambda 1lmbda>=0] comes from?

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[frank marsh]

Markov inequality implies the Chebyshev inequality, for one, very useful tool in probablity theory

the bottom inequality follows from monotonicity of integral

there's nothing special to this proof, just definitions

revise them

[somber urchin]

How come is X 1x>=lbd >= lbd 1x>=lbd??

X is a positive random variable, it can be a lot of high values

and you tell me that even so it is always bigger than any lambda chosen?

Can't I just pick a high enough lbd to get over the higehst value of X?

[frank marsh]

what are you even asking

[somber urchin]

Literally that

[frank marsh]

you might understand that gibberish but I don't

now put some effort into it

[frank marsh]

because of this

$$ {X\geqslant \lambda} $$

do you understand what this set is?

[swift sky]

aL

[somber urchin]

Set of all points in the sample space satisfying X(x) >= lbd

[frank marsh]

good

so if X(w) < lambda, then the product is zero, yes?

[somber urchin]

Yes

[frank marsh]

hence we only care about those trajectories on which X(w) >= lambda

[somber urchin]

Yes

[frank marsh]

and that's all

[somber urchin]

??

[frank marsh]

that's why the inequality holds

[somber urchin]

ooooooooh

I see now

[frank marsh]

you are multiplying by 1

[somber urchin]

I mistook the X below as a small x

[frank marsh]

$$ X(\omega) \geqslant \lambda \Rightarrow I_{{X\geqslant \lambda}}(\omega) = 1 $$

[somber urchin]

No need for further explanations guys

[swift sky]

aL

[somber urchin]

I get it now

[obsidian bronze]

Basically $\forall \omega \in \Omega, X(\omega) \geq X(\omega)1_{X \geq \lambda}(\omega) \geq \lambda 1_{X \geq \lambda}(\omega)$ you can check that’s it’s true and so using the property of the expectation you have the result

[swift sky]

😑 rotoR

[obsidian bronze]

Ah okay good

[somber urchin]

+close