Chebyshev's Theorem

• If a random variable has a small variance or standard deviation, we would expect most of the values to be grouped around the mean
• A large variance indicates a greater variability, so the area of distribution should be spread out more.

  Figure 4.2: Variability of continuous observations about the mean.

  

  Figure 4.3: Variability of discrete observations about the: mean.

  

• Theorem 4.11:
  
• Example 4.22: A random variable $X$ has a mean $\mu= 8$, a variance $\sigma^2=9$, and an unknown probability distribution. Find
• $P(-4<X<20)$
  $$P(\mu-k\sigma <X < \mu + k\sigma) \geq 1-\frac{1}{k^2}$$
  $$P(-4<X<20)=P(8-4*3 <X < 8+4*3) \geq 1-\frac{1}{4^2}=\frac{15}{16}$$

• $P(\vert X-8\vert \geq 6)$
  $$P(\vert X-8\vert \geq 6)=1-P(\vert X-8\vert < 6)=1-P(-6<X-8<6)$$
  $$=1-P(8-6<X<6+8)=1-P(8-2*3<X<8+2*3) \leq \frac{1}{2^2}=\frac{1}{4}$$

• The Chebyshev inequality is a useful tool as well as a relation that connects the variance of a distribution with the intuitive notation of dispersion in a distribution.
• For any population or sample, this provides that the minimum probability of the data within $k\sigma$ from the mean $\mu$ is $1-\frac{1}{k^2}$.
• The use of Chebyshev's theorem;
  • holds for any distribution of observations
  • gives a lower bound only
  • is suitable to situations where the form of the distribution is unknown (a distribution-free result)