Central limit theorem

Question

I'm stuck on this idea from my lecture notes:

Using the Central Limit Theorem, $$Y^{(n)}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{Y_i-\mu}{\sigma}$$ Then $$\lim_{n\rightarrow\infty}P(Y^{(n)}\leq u)=\int_{-\infty}^{u}e^{\frac{-t^{2}}{2}} \, dt$$

Surely since it is being approximated by the standardised normal approximation it needs to normalized and needs a constant $\frac{1}{\sqrt{2\pi}}$? If not can someone explain why?

Thanks

Answer 1

It appears that there is a typo. It should say $$ \lim_{n\rightarrow\infty}P(Y^{(n)}\leq u)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{u}e^{\frac{-t^{2}}{2}} \, dt $$