A Central Limit Theorem simple example

Question

A disscusion in the book: Let $(X_n)_{n=1}^\infty$ a sequence of i.i.d random variables such that $\mathbb{E}[X_n]=60, \operatorname{Var}[X_n]=25$. Let $S_N= \sum_{i=1}^NX_i$. By the central limit theorem we get $\frac{1}{5\sqrt{N}}\overset{D}{\rightarrow}N(0,1)$. I don't understand why does the next equality hold: $$ \\ \mathbb{P}[55<{1\over N}S_N<65]=\mathbb{P}[-\sqrt N<{S_N-60N\over 5 \sqrt N}<\sqrt N] \ $$

Answer 1

The following statements are equivalent:

• $55<\frac1{N}S_N<65$
• $55N<S_N<65N$
• $-5N<S_N-60N<5N$
• $-N<\frac{S_N-60N}5<N$
• $\frac{-N}{\sqrt N}<\frac{S_N-60N}{5\sqrt N}<\frac{-N}{\sqrt N}$
• $-\sqrt N<\frac{S_N-60N}{5\sqrt N}<\sqrt N$