EC484_Econometrics_Analysis

# Chebyshev Inequality

The Chebyshev Inequality provides a bound on the probability that a random variable deviates from its mean by more than a specified number of standard deviations. It is particularly useful because it applies to any probability distribution with a finite mean and variance, regardless of the distribution’s shape.

The statement of Chebyshev Inequality is:

Theorem For any random variable $z_n$ and constant $\delta >0$, it holds

$$\mathrm{Pr}(|z_n-\mathbb{E}[z_n]>\delta )\le \frac{\text{Var}(z_n)}{{\delta }^2}$$

By setting $z_n=\bar{y}$, we could have $\mathbb{E}[z_n]=\mathbb{E}[y]$, and $\text{Var}(z_n)=\frac{{\sigma }^2}{n}$.

Why we have $\text{Var}(z_n)=\frac{{\sigma }^2}{n}$? Because from the definition of variance, we have:

\text{Var}(\bar{y}) = \text{Var}\left( \frac{1}{n} \sum_{i=1}^{n} y_{i} \right) = \frac{1}{n^2} \sum_{i=1}^{n} \text{Var}(y_{i}) = \frac{n \sigma^2}{n^2} = \frac{\sigma^2}{n}$$

Thus we could use it to derive Weak Law of Large Numbers

Proof

Pick arbitrary $\delta >0$, let $F_n(u)$ be distribution function of $z_n-\mathbb{E}[z_n]$, then:

$$\mathrm{Pr}(|z_n-\mathbb{E}[z_n]|>\delta )=\mathrm{Pr}((z_n-\mathbb{E}[z_n]{)}^2>{\delta }^2)$$

We let $u=z_n-\mathbb{E}[z_n]$, then:

$$={\int }_{u^2>{\delta }^2}d{F}_n(u)$$