Central Limit Theorem for product of random variables

Question

How would one approximate the following using the Central Limit Theorem:

Let $(X_n)_{n \in \mathbb{N}}$ be independent random variables that are uniformily distributed over [0, 1].

Let $Y_n := (\prod^n_{i=1} X_i)^{\frac{1}{n}}$

Find: $\mathbb{P}(Y_{50} \geq \mathbb{E}(Y_{50}))$

Thanks in advance :)

Answer 1

Let's spell out the hint:$$\begin{align}X_n&\sim U(0,\,1)\\\implies\ln X_n&\sim\operatorname{Exp}(1)\\\implies(\Bbb E\ln X_n,\,\operatorname{Var}\ln X_n)&=1\\\implies(\Bbb E\ln Y_n,\,\operatorname{Var}\ln Y_n)&=(1,\,\tfrac1n).\end{align}$$For large enough $n$ to use the CLT,$$\begin{align}\ln Y_n&\approx N(1,\,\tfrac1n)\\\implies Y_n&\sim\operatorname{Lognormal}(1,\,\tfrac1n)\\\implies\ln\Bbb EY_n&=1+\frac{1}{2n}\\\implies P(\ln Y_n\ge\ln\Bbb EY_n)&=\Phi\left(-\frac{1}{2\sqrt{n}}\right).\end{align}$$

Answer 2

By taking the $\log Y_n$ ones gets $\frac{1}{n} \sum^n_{i=1} \log(X_i)$ which can then be approximated by the Central Limit Theorem.

Thanks to anomaly