Limit theorems

Description:

• Markov’s inequality
• Chebyshev’s inequality
• Central limit theorem
• Law of large number

Practical interpretation:

• Suppose a random variable $X$ has mean $\mu$ and standard deviation $\sigma$. Then:
• Sum of observations, $S_n=X_1+\cdot \cdot \cdot +X_n$ has mean $n.\mu$ and standard deviation $\sigma \sqrt{n}$
• Sample mean, ${\bar{X}}_n=\frac{X_1+\cdot \cdot \cdot +X_n}{n}$ has mean $\mu$ and standard deviation $\frac{\sigma }{\sqrt{n}}$
• If $X$ is normally distributed, then $X_n$ and $S_n$ are normally distributed for any $n$
• If $X$ is not normally distributed, $X_n$ and $S_n$ are approximately normally distributed provided $n$ is sufficiently large.