I am stuck with the following problem, where I am asked to prove/disprove the following hypothesis:
Is $\mathrm{E}\{e^{\max_i X_i}\} = \mathrm{E}\{\max\limits_i e^{X_i}\}$,
where the $X_i$'s are dependent random variables.
It seems intuitively true, but I trying to write down a proof. Anybody has any hints ?
On
We can deal with equality of expected values by showing the two random variables are equal.
$$ e^{\max_i X_i} = \max_i\{e^{X_i}\}. \tag1 $$ No expected values are mentioned on the line above.
If two random variables are really the same random variable, then their two expected values are the same.
You'd have a different situation if you were asking about $\max_i\{\operatorname{E}(X_i)\}$. That would often be smaller than the expected value of the random variable in $(1)$.
Hint: $e^x$ is a strictly increasing function so it preserves order among variables if you have multiple variables. Thus $\max_i(e^{X_i}) = e^{\max_i X_i}$