Name: Lec 24｜MIT 6.042J コンピュータサイエンスのための数学 2010年秋学期 (Lec 24 | MIT 6.042J Mathematics for Computer Science, Fall 2010)
Uploaded: 2021-01-14T06:23:10.000Z
Duration: 1 h 23 min 23 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

基礎受動態

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PROFESSOR: So today, we're going to talk about the probability

that a random variable deviates by a certain amount

Now, we've seen examples where a random variable

is very unlikely to deviate much from its expectation.

heads, very unlikely to wind up with 25 or fewer

We've also seen examples of distributions

where you are very likely to be far from your expectation,

for example, that problem when we had the communications

channel, and we were measuring the latency of a packet

So you're very likely to deviate a lot from your expectation

And we saw how that gave us some feel for the likelihood

high variance meaning you're more likely to deviate

Today, we're going to develop specific tools

for bounding or limiting the probability you

deviate by a specified amount from the expectation.

And the first tool is known as Markov's theorem.

Markov's theorem says that if the random variable is always

non-negative, then it is unlikely to greatly exceed

In particular, if R is a non-negative random variable,

then for all x bigger than 0, the probability

that R is at least x is at most the expected value of R,

So in other words, if R is never negative-- for example,

Then the probability R is large will be a small number.

Because I'll have a small number over a big number.

So it says that you are unlikely to greatly exceed

Now, from the theorem of total expectation

by looking at two cases-- the case when R is at least x,

That's from the theorem of total expectation.

Take the expected value there times the probability

of this case happening plus the case when R is less than x.

OK, now since R is non-negative, this is at least 0.

Because I'm taking the expected value of R

So that means that the expected value of R

R is greater than x, R is greater or equal to x.

And now I can get the theorem by just dividing by x.

I'm less than or equal to the expected value

But it's going to have amazing consequences

that we're going to build up through a series of results

Any questions about Markov's theorem and the proof?

All right, there's a simple corollary, which is useful.

Again, if R is a non-negative random variable,

then for all c bigger than 0, the probability

that R is at least c times its expected value is at most 1

So the probability you're twice your expected value

We just set x to be equal to c times the expected

So I just plug in x is c times the expected value of R.

And I get expected value of R over c times the expected value

So you just plug in that value in Markov's theorem,

Let's let R be the weight of a random person

And I don't know what the distribution of weights

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Lec 24｜MIT 6.042J コンピュータサイエンスのための数学 2010年秋学期 (Lec 24 | MIT 6.042J Mathematics for Computer Science, Fall 2010)

negative

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