Let represent the mutual information between random variables X and Y.

Here I give example one for the following inequalities.

** Example 1**: For

Let X, Y be independent uniformly distributed random bits and Z=X Y. Then,

LHS =

= =1

RHS = 0

Hence, .

* Example 2*: For

Let X=Y=Z, X is a uniformly distributed random bit. Then, it can be shown that

LHS = 0

RHS= 1