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#1 2013-02-20 09:43:20

White_Owl
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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?

#2 2013-02-20 09:45:48

anonimnystefy
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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.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#3 2013-02-20 10:31:02

White_Owl
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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 (2013-02-20 10:31:37)

#4 2013-02-20 10:46:10

anonimnystefy
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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 .


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#5 2013-02-20 11:02:16

White_Owl
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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)???

#6 2013-02-20 11:33:23

anonimnystefy
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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.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#7 2013-02-20 11:46:42

White_Owl
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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...

#8 2013-02-22 07:09:58

White_Owl
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Re: Chebyshev's Inequality

Well? Does anyone have an answer?

#9 2013-02-22 14:47:54

Avon
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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.

#10 2013-02-23 16:14:44

White_Owl
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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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