Let $X_1,X_2,…,X_n$ be i.i.d $Unif[0,1]$ random variables. Show that $Y=(\prod_{i=1}^nX_i)^{1/n} \rightarrow c$, and find the constant c.
My Attempt
First, we take the log of the $Y$ so that we can bring down the exponent.
$$\ln(Y)=\frac{1}{n}\ln(\prod_{i=1}^nX_i)=\frac{1}{n}\sum^n_{i=1}\ln(X_i)$$
By Strong Law of Large Numbers, we have
$$\mathbb{P}(\lim_{n\rightarrow\infty}\frac{1}{n}\sum^n_{i=1}X_i=\mu)=1$$
So we know that the limit converges to $\mu$ with probability 1, where $\mu$ is the finite mean. Now we just need to find the expectation of $X$, but that is where I am stuck. We know the expectation of a uniform distribution on $[0,1]$ is $\mathbb{E}[X]=\frac{1-0}{2}=\frac{1}{2}$. But surely, this cannot be the answer.
Best Answer
$E(\ln X_1)=\int_0^{1} \ln x dx=x\ln x-x|_0^{1}=-1$.
[$Eg(X_1)=\int_0^{1} g(x)f(x)dx$ and the density function $f(x)$ is $1$ for $0 <x<1$].