from the form $P \left\{ \frac{S_n - n\mu}{\sigma\sqrt{n}} < \beta \right\} \to \mathfrak{N}(\beta)$ to the form $P \{ |S_n - n\mu| < \beta \sigma \sqrt{n} \} \approx \mathfrak{N(\beta)} - \mathfrak{N(-\beta)}$?
How to convert the Central Limit Theorem
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probability-theory