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#1 2013-02-19 10:43:20

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Chebyshev's Inequality

I do not understand it.
Well, I kinda understand its meaning, but I am not sure how to read it or calculate it on a real numbers.


For example, if we have a pmf  table:

Now, how to calculate it for lets say k=10? or for k=100?
What goes for X in the inequality and what is the result of Pr() function?

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#2 2013-02-19 10:45:48

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Chebyshev's Inequality

That inequality means that the probability of the difference between X and mu being greater or equal to k*sigma is less than or equal to 1/k^2.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#3 2013-02-19 11:31:02

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Chebyshev's Inequality

Yes, I read the definition, but it is not enough, I need a practical example.
Definition also says that it is true for all k>0.
So, if we have k=10:


Which X should I put here to see the truth of this inequality?

Last edited by White_Owl (2013-02-19 11:31:37)

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#4 2013-02-19 11:46:10

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Chebyshev's Inequality

You need to find the probability that |X-3.2|>=18.3 . You can do that by summind the probabilities of all values of X for which that is hreater than 18.3 (which is in this case 0). And you probably know that 0<=0.01 .


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#5 2013-02-19 12:02:16

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Chebyshev's Inequality

Ok, then lets take the k=2.


In that case we have two X which satisfy the inequality in parenthesis: 7 and 8.
Now what is the values of

Can you tell me the values for a and b?
What is the value of Pr(TRUE)???

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#6 2013-02-19 12:33:23

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Chebyshev's Inequality

They are both 1. But if you multiply 1 by the probability that X is in fact 7 (or 8), and then sum over all values of X, you will get the probability.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#7 2013-02-19 12:46:42

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Chebyshev's Inequality

I am sorry, I do not understand your answer.
You said a=b=1? Where did 1 come from?
What exactly should I sum and why?
I am now even more confused...

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#8 2013-02-21 08:09:58

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Chebyshev's Inequality

Well? Does anyone have an answer?

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#9 2013-02-21 15:47:54

Avon
Member
Registered: 2007-06-28
Posts: 80

Re: Chebyshev's Inequality

White_Owl,

Suppose you wanted to find

Hopefully it is obvious that

exactly when
or
or
,

so

Similarly, you have observed that

exactly when
or
,

so


What anonimnystefy is referring to in post #6 is that


where
if
is true and 0 if it is false.
This is also equal to


I hope this helps.

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#10 2013-02-22 17:14:44

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Chebyshev's Inequality

Oh! So inequality inside parenthesis in the probability arithmetic means sum of probabilities of all events which satisfy the inequality?
Well, it is kinda strange and unintuitive, but ok. I guess there is some logic in that and I would have to live with it...

But why not just write something like:


I think this would be more clear. Don't you think?

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