User10405, I didn't mean to offend you by the homework comment, but proving that the sum of independent Bernoullis is a Binomial is exactly a homework problem I assigned the statistics class (Mathematical Statistics 1) I teach recently. Apologies for the "useless prejudice."
I'm confused, Macro's comment seemed appropriate, yet:
Macro: This is not a bunch of Bernoulli trials and a binomial distribution does not apply.
But in the original statement:
Let $X_1,X_2,...,X_n$ be a fixed number of Bernoulli random variables
See http://en.wikipedia.org/wiki/Bernoulli_random_variable for verification that the sum of Bernoullis is indeed a Binomial, and since $avg$ is just the sum divided by $n$, it will be a scaled Binomial.
As far as $f=max$ goes, since each of the X 's is a Bernoulli, i.e. 0 or 1, the max also has to be 0 or 1. It will be 0 only if all of the X 's are 0. So treat it as a Bernoulli($p∗$) where $p∗=1−P(X_1=X_2=...=X_n=0)$ .
An assumption of independence among the $X$'s makes $p∗$ much easier to calculate.
I get the feeling that this answer still won't be what you're looking for, but without more clarification, I don't see how we can help further.