Question: Let (left(X_{1}, ldots, X_{n}ight)) be iid random variables with (sigma^{2}

Let $\left(X_{1}, \ldots, X_{n}ight)$ be iid random variables with $\sigma^{2}<\infty$. We know from Khinchin's WLLN that

$\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i} \xrightarrow{\mathrm{p}} u$.

(a) Find a functional relationship between $n, \sigma$, and $\varepsilon$ such that

$P(\bar{x} \in(\mu-\varepsilon, \mu+\varepsilon)) \geq .99$.

(b) For what values of $n$ and $\sigma$ will an outcome of $\bar{X}$ be within $\mp .1$ of $\mu$ with probability $\geq .99$ ? Graph this relationship between the values of $n$ and $\sigma$.

(c) If $\sigma=1$, what is the value of $n$ that will ensure that the outcome of $\bar{X}$ will be within $\mp .1$ of $\mu$ with probability $\geq .99$ ?

(d) If $\sigma=1$, and the $X_{i}$ 's are normally distributed, what is the value of $n$ that will ensure that the outcome of $\bar{X}$ will be within $\mp .1$ of $\mu$ with probability $=.99$ ?

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