Maybe it is impossible. But how with this problem:
In the rolling of a single fair die, Suppose X be the outcome.
Then E(X)=3.5
and Var(X)=35/12
Let we want to compute P(X>=5) by the usual Chebyshev inequality,
we obtain
P(X>=5) =5 or X =1.5) 1 How come?
The estimate provided by the Chebyshev inequality does not give any information here. Since the probability of any event is bounded by 1 anyway, it is also bounded by 1.296296 (and by 1.1, by 17, and by any number larger or equal 1).
1) In answer to the title question: probability is a function that maps to the interval [0, 1]. Values less than zero and greater than 1 are not probabilities.
2) Chebyshev's inequality (No more than 1/k2 of the distribution's values can be more than k standard deviations away from the mean) states the percentage of observations that we would expect to find within a given number of standard deviations. It does not purport to calculate the probabilities per se.
3) Ignoring everything else, your equation is an inequality which states, in essence P(X >=5)
Chebyshev's inequality is useful when k >1. In your example, k < 1 and the inequality is trivial as probability is
Though it is not specified in the statement of the inequality that k >1, it has to be. otherwise we may may end up with these kinds of answers
On real based phenomena, probability is conserved. This means that the horizon of different responses of a phenomenon must sum up to 1.
Interestingly, it has been proposed that you can get probabilities greater than one and less than zero in quantum mechanics.
https://physics.stackexchange.com/questions/11049/does-the-extended-probability-ensemble-interpretation-of-quantum-mechanics-make
This has nothing to do with Chebyshev inequality.
Inequalities only provide bounds and not values.By definition probability cannot assume a value less than 0 or greater than 1.
Chebyshev inequality only give us an upper bound for the probability. In this case you have P(|X-3.5|>=1.5) =1.5)
The axiomatic definition of Probability states one of the axioms as, for any event A, 0
The value of probability always lies between 0 and 1, so that it never cannot be greater than 1
The definition of probability is composed in such a way that the value of probability lies between -1 and +1. It can not be more than +1 and less than -1.
In fact, usually probability is even defined to be from the interval [0,1]. A probability measure is a nonnegative measure (on a measurable space (Omega,F)) which assigns the value 1 to Omega.
Dhritikesh Chakrabarthy is confusing the numerical value of correlation coefficient with that of probability. The extreme values of probability are 0 and 1 and not -1 and 1 as mentioned by Chakrabarthy.
I am sorry for the printing error committed in my earlier answer on this question.
The correct answer will be as follows:
"The definition of probability is composed in such a way that the value of probability lies between 0 and +1. It can not be more than 1 and less than 0."
I am ever grateful to Natesan Thillaigovindan for pointing out the error committed by me in my earlier answer.
Lot of thanks to him.
Also, I tender apology to all the readers for this error.
The probability is always between 0 and 1. Your démonstration is true. The problem is that you have give a majoration of the probability in question and not its value. We say Chebyshev Inequality not Chebyshev equality
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability