Chapter 4 Entropy

4.1 Definition

Let $X$ be a random variable and $P_X(x)$ be its probability density function (pdf). The entropy $H(X)$ can be interpreted sa measure of the uncertainty of $X$ and is defined in the discrete case as follows: $$\begin{equation} H(X) = -\sum_{x \in X}{P_X(x)\log{P_X(x)}}. \label{eq:H} \end{equation}$$

If the $\log$ is taken to base two, then the unit of $H$ is the (binary digit). We employ the natural logarithm which implies the unit in (natural unit of information).

4.2 Nonlinear Coupling

4.2.1 Simulated Systems

4.2.2 Equity-Commodities Relationship

4.3 Efficiency and Bubbles: A Case Study in the Crypto and Equity Markets