这是一份umass麻省大学 STAT 501作业代写的成功案例
Let $X_{1}, X_{2}, X_{3}, \ldots$ be a sequence of mutually independent random variables, each with the same distribution as $X$. Two related sequences are of interest:
- The sequence of running sums:
$$
S_{m}=\sum_{i=1}^{m} X_{i} \text { for } m=1,2,3, \ldots
$$
- The sequence of running averages:
$$
\bar{X}{m}=\frac{1}{m} \sum{i=1}^{m} X_{i} \text { for } m=1,2,3, \ldots
$$
STAT501 COURSE NOTES ：
Using complements, it is sufficient to demonstrate that
$$
\lim {m \rightarrow \infty} P\left(\mu-\epsilon<\bar{X}{m}<\mu+\epsilon\right)=1
$$
Since $S D\left(\bar{X}{m}\right)=\sigma / \sqrt{m}$, if $\epsilon=k \sigma / \sqrt{m}$ (correspondingly, $k=\epsilon \sqrt{m} / \sigma$ ) is substituted into the Chebyshev inequality, then we get the following lower bound on probability: $$ P\left(\mu-\epsilon<\bar{X}{m}<\mu+\epsilon\right) \geq 1-\frac{1}{k^{2}}=1-\frac{\sigma^{2}}{\epsilon^{2} m}
$$
As $m \rightarrow \infty$, the quantity on the right in (ii) approaches 1 , implying that the limit in (i) is greater than or equal to 1 . Since probabilities must be between 0 and 1 , the limit must be exactly $1 .$