Hello everyone!
The problem is:
Suppose that the probability that a text contains 10 or more instances of the word "president" is 1/5. Provide a lower bound on the expected number of instances of this word
The expected number of instances is:
$\displaystyle E(n)=\sum_{r=0}^{\infty} r p(r)$
This is obviously minimised subject to the constraint if $\displaystyle p(0)=0.8$, $\displaystyle p(10)=0.2$ and $\displaystyle p(r)=0$ for all other $\displaystyle r$.
So:
$\displaystyle E(n) \ge 2$
and the bound is tight.
RonL