Let

$$

$$be a finite set of symbols. Let

$$

$$be a pmf on

$$

$$with entropy

$$

$($

$$

$)$

$.$

$$Let

$$

:

$$

$->$

$$

$$be the code lengths of a prefix code on

$$

$.$

$$We think of

$$

$$as a random

variable on

$$

$.$

$$Show that

$$

$$

$2$

$$

$>=$

$2$

$$

$($

$$

$)$

$.$