(strong) law of large numbers

Once more we will begin with the epsilon-delta technique.

In the probability theory there is a theorem called "(strong) law of large numbers". It is one of most famous theorems in the probability or statistics.

Random variables $X_1,X_2,\cdots ,X_n$ are independent identically distributed(i. i. d.). $E(X_i)=\mu<\infty$. Then,
\[ Y_n=\frac{1}{n}\sum_{i=1}^nX_i\rightarrow\mu\quad (n\rightarrow\infty)\quad a.s. \]
In other words,
\[  P\left\{ \lim_{n\rightarrow\infty}Y_n=\mu \right\}=1 \]

Law of large numbers means the average of sample data will converge to the expectation value of each random variable (which all will become same),  when sample data increase infinitely.
In general, law of large numbers indicates the strong law.

In next post, I will explain how applied the epsilon-delta technique is.