The Notebook

**Chebyshev Inequality**

Let

Then, the Chebyshev inequality states that

**P(|X - \mu| \geq t) \leq \frac{\sigma^2}{t^2}**, for **t > 0**.

Let A denotes the event

**\sigma^2 = E[(X - \mu)^2|A]P(A) + E[(X - \mu)^2|A^{c}]P(A^{c})**

**\sigma^2 \geq E[(X - \mu)^2|A]P(A)**

since

**0 \leq P(A^{c}) \leq 1 ** and **E[(X - \mu)^2|A^{c}] \geq 0**.

So, whenever

finally, by using third equation, can say that,

**P(|X - \mu| \geq t) \leq \frac{\sigma^2}{t^2}**

**Weak Law of Large Numbers**

Let

Let

**P(|\frac{S_n}{n} - \mu| \geq t) \rightarrow 0**, for **t > 0** as **n \rightarrow \infty**.

Since

**\mathrm{Var[S_n]} = n \sigma^2**

and,

**\mathrm{Var[\frac{S_n}{n}]} = \frac{\sigma^2}{n}**

By Chebyshev's inequality, for any

**P(|\frac{S_n}{n} - \mu| \geq t) \leq \frac{\sigma^{2}}{nt^{2}}**

Thus, for fixed

**P(|\frac{S_n}{n} - \mu| \geq t) \rightarrow 0**

as

**P(|\frac{S_n}{n} - \mu| < t) \rightarrow 1**

**Appendix**

** ****\mathrm{Var[\frac{S_n}{n}]} = \frac{\sigma^2}{n}**

**\mathrm{Var}[\frac{S_n}{n}] = \mathrm{Var}[\frac{X_1}{n}] + \mathrm{Var}[\frac{X_2}{n}] + \dots + \mathrm{Var}[\frac{X_n}{n}]**

**\frac{1}{n^2}(\mathrm{Var}[X_1] + \mathrm{Var}[X_2] + \dots + \mathrm{Var}[X_n])**

** \frac{1}{n^2} \times n \times \sigma^2 = \frac{\sigma^2}{n} **

- http://www.ams.sunysb.edu/~jsbm/courses/311/cheby.pdf
- https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter8.pdf
- https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter6.pdf
- http://textofvideo.nptel.ac.in/117104129/lec9.pdf