What is difference between taking the lower bound and lower limit from Chebyshev's Theorem?
If P(|Y- mean|) <= 1 - (1/r^2)
Do we have to consider 1 - (1/r^2) as the lower bound?
To understand the difference between lower bound and lower limit in Chebyshev's Theorem, let's first establish what Chebyshev's Theorem states. Chebyshev's Theorem is a mathematical inequality that provides bounds on the probability of a random variable deviating from its mean value.
The theorem states that for any random variable Y with mean μ and standard deviation σ, the probability that the absolute difference between Y and its mean |Y - μ| is greater than or equal to some positive value kσ, is at most 1/k^2.
Now, let's clarify the terms "lower bound" and "lower limit" in the context of Chebyshev's Theorem.
The lower bound refers to the minimum probability specified by the theorem. In the case of Chebyshev's Theorem, the lower bound is given by 1 - (1/k^2). This means that the probability of the absolute difference between Y and its mean being greater than or equal to kσ is at least this value.
On the other hand, the lower limit is not a term used in Chebyshev's Theorem. It seems you might be referring to the lower limit of |Y - μ| that is applicable to specific values of k, rather than the general bound provided by the theorem.
In regards to the expression P(|Y - μ| <= 1 - (1/r^2)), it does not represent the lower bound specified by Chebyshev's Theorem. Instead, it appears to be an attempt to define a probability based on a specific value for the absolute difference |Y - μ|. The expression itself is not a valid mathematical representation, as the right-hand side should be in the form of a multiple of the standard deviation (such as kσ, where σ represents the standard deviation of Y).
To correctly apply Chebyshev's Theorem, you would need to specify a positive value k and use the appropriate expression provided by the theorem. This expression will give you the lower bound on the probability of the absolute difference between Y and its mean being greater than or equal to kσ.