I am studying Chernoff bounds lately but I am not clear how this is better than cumulative distribution function for binomial distribution.
For example, let's say that
p = 0.3 and
n = 3. I am interested in finding $Pr(X \geq 2)$.
If I do cumulative distribution, this will be
P(X = 2) + P(X = 3), and the answer comes out to be
But when I apply Chernoff bound for $Pr(X \geq 2)$, I get an upper bound of
0.6. How is Chernoff bound any useful then? Why not just use cumulative distribution always as it seems to give better bounds. What am I missing?
(I used this to find the Chernoff bound)