91Ó°ÊÓ

we have to tell about pros and cons of Chebyshev's rule.

Pros:

1. Chebyshev’s Theorem require only a finite second moment.
2. It is tremendously useful to establish weak laws of large numbers (mostly because of point 1 above).
3. Chebyshev’s rule applied to a series of random variables does not require independence of the elements of the sequence.

• Cons:

1. It never provides a sharp bound on the probability. For example, Chebyshev’s rule will not guarantee that any of the mass/density functions have a positive probability having 1 $\mathrm{σ}and\mathrm{μ}$. But the Normal distributions have about 2/3 of the probability lying between and $\mathrm{μ}-\mathrm{σ}\text{and}\mathrm{μ}+\mathrm{σ}$
2. Sharpening the bound require stronger assumptions about the random variables in question.
3. Chebyshev’s rule is more useful as a tool in theoretical statistics and probability theory than in data analysis.