Given the sequence $(X_n), n=1,2,... $, of iid exponential random variables with parameter $1$, define:
$$ M_n := \max \left\{ X_1, \frac{X_1+X_2}{2}, ...,\frac{X_1+\dots+X_n}{n} \right\} $$ I want to calculate $\mathbb{E}(M_n)$.
Define for $k = 1, 2, ..., n$
$$Y_k := \frac{X_1 + ... + X_k}{k}$$
Then we have $E[Y_k] = 1$
We can rewrite:
$$M_n = \sum_{k=1}^{n} Y_k 1_{A_{n,k}}$$
where $A_{n,k}$ is the event that $Y_k = \max\{Y_1, ..., Y_n\}$
I was thinking that $E[M_n] = 1$ using independence, but then again, intuitively, the larger $Y_k$ is, the greater is the probability that it is the maximum.
Why rigorously is it that $\sigma(Y_k)$ and $\sigma(A_{n,k})$ are not independent?
I know that $\sigma(Y_k) = \{Y_k^{-1}(B) | B \in \mathscr B\}$ but have no clue as to what Borel set to use.