Q. 3.96 - Elementary Statistics

Question

Discuss the pros and cons of Chebyshev's rule.

Step-by-Step Solution

Answer

pros and cons are being explained in description.

Step 1: Givem information

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

Step 2: Explanation

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 $\sigma and \mu$. But the Normal distributions have about 2/3 of the probability lying between  and $\mu -\sigma \text{ and }\mu +\sigma$
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.