What would you like to understand?

Joint entropy $H(X,Y)$ is not equal to $H(X) + H(Y)$ because adding the marginal entropies double-counts the mutual information I(X;Y) shared between the variables. The correct relationship is $H(X,Y) = H(X) + H(Y) - I$(X;Y).

Conditions: X and Y are random variables that may share information (i.e., they are not necessarily independent).

Joint entropy $H(X,Y)$ is not equal to $H(X) + H(Y)$ because adding the marginal entropies double-counts the mutual information I(X;Y) shared between the variables. The correct relationship is $H(X,Y) = H(X) + H(Y) - I$(X;Y).

Conditions: X and Y are random variables that may share information (i.e., they are not necessarily independent).

Conditional entropy H(X|Y) is bounded by $0 \le H$(X|Y) ≤ $H(X)$ because learning the value of Y provides information that can only reduce or leave unchanged the average uncertainty about X; it cannot increase it on average.

Conditions: X and Y are discrete random variables.

Conditional entropy H(X|Y) is bounded by $0 \le H$(X|Y) ≤ $H(X)$ because learning the value of Y provides information that can only reduce or leave unchanged the average uncertainty about X; it cannot increase it on average.

Conditions: X and Y are discrete random variables.