Name: Lec 24 | 麻省理工學院 6.042J 計算機科學數學，2010 年秋季。 (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: 【看影片學英語】數萬部 YouTube 影片，搭配英漢字典即點即查，輕鬆掌握單字發音與用法，長久累積看電影不必再看字幕。

基礎被動語態

The following content is provided under a Creative

Your support will help MIT OpenCourseWare

continue to offer high quality educational resources for free.

To make a donation or view additional materials

from hundreds of MIT courses, visit MIT OpenCourseWare

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

字幕列表影片播放

Lec 24 | 麻省理工學院 6.042J 計算機科學數學，2010 年秋季。 (Lec 24 | MIT 6.042J Mathematics for Computer Science, Fall 2010)

negative

expect

random

audience