Let $X_1, \dots , X_n$ i.i.d. uniformly distributed random variables with $f(x) = 1_{(0,1)}(x)$, $x \in \mathbb{R}$. Let $\Pi_n = (X_1 \dots X_n)^\frac{1}{n}$ and $M_n = \max \{ X_1, \dots , X_n \}$. Prove the following:
- $E(X_1 \dots X_n)^\frac{1}{n} \leq (EX_1 \dots EX_n)^\frac{1}{n}$
- $P(\Pi_n \leq M_n) = 1$.
Already I showed that $E \Pi_n = (1 - \frac{1}{n+1})^n = (\frac{n}{n+1})^n$ and $(EX_1\dots EX_n)^\frac{1}{n} = \frac{1}{2}$. So now my plan was to use induction to prove the first statement. It clearly holds for $n=1$. Next is to show that $(\frac{n+1}{n+2})^{n+1} \leq \frac{1}{2}$, using $(\frac{n}{n+1})^n \leq \frac{1}{2}$. But I don't see how to prove that, hoping this is the right approach for the problem.
For the second part I was thinking to say $(\Pi_{i=1}^n x_i)^\frac{1}{n} \leq \frac{1}{n} \sum_{i=1}^n x_i \leq \max\{x_1, \dots , x_n\}$. Would that suffice?