ThmDex – An index of mathematical definitions, results, and conjectures.

Mutual information of a simple random variable with respect to itself

Formulation 0

Let $X : \Omega \to \mathcal{X}$ be a D5723: Simple random variable.
Let $a \in (0, \infty) \setminus \{ 1 \}$ be a D5407: Positive real number.

Then \begin{equation} I_a(X ; X) = H_a(X) \end{equation}

Proofs

Proof 0

Let $X : \Omega \to \mathcal{X}$ be a D5723: Simple random variable.
Let $a \in (0, \infty) \setminus \{ 1 \}$ be a D5407: Positive real number.

Using results

(i)	R4843: Conditional entropy formula for simple mutual information
(ii)	R4846: Conditional simple entropy with respect to itself

we have \begin{equation} I_a(X ; X) = H_a(X) - H_a(X \mid X) = H_a(X) - 0 = H_a(X) \end{equation} $\square$