Entropy of an ensemble

The entropy of an ensemble, $X=(x,A_x,P_x)$, is defined to be the average Shannon information content over all outcomes:

Properties of entropy:

• [$H(X)\ge ?$ ] with equality iff [...].
• Entropy is maximized if [something about the outcomes is true].
• The Entropy is less than [some opperation applied to] the number of outcomes.

The last two points can be expressed as:

Proof on the back side.