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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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#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?