Considering the bin case as described in the book and the lectures, we could write the probability of

being close to

:

can be written excatly since

is binomaily distributed with parameters

and

. Now, the problem is that that expression has

in it which is unknown. But we can find the value of

for which

has its maximum, and it turns out to be

.

Now if we plug that in the expression we get the bound with same properties as Hoeffding's (valid for all

's,

's and

's).

Now, my view is that that bound is the tightest possible, tighter than the Hoeffding's. Am I correct? It could be (perhaps?) used in futher analysis, but the major donwside is that is is not so nice nor elegant to work with. Other opinions?

Thanks!