State the LLN operationally

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Problem

If $X_1,X_2,\ldots$ are i.i.d. $\operatorname{Bernoulli}(0.3)$ and $\bar X_n=n^{-1}\sum_iX_i$ , state the weak law of large numbers for this case. What does it imply about $P(|\bar X_n-0.3|>0.05)$ as $n\to\infty$ ?

Reveal answer or reference solution

The weak law says $\bar X_n\xrightarrow{p}0.3$ . Equivalently, for every $\varepsilon>0$ ,

P(|\bar X_n-0.3|>\varepsilon)\to0

Taking $\varepsilon=0.05$ gives $P(|\bar X_n-0.3|>0.05)\to0$ .

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State the LLN operationally

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