How can you calculate the entropy of a discrete random variable using its probability mass function?

Calculating Entropy of a Discrete Random Variable

Entropy is a measure of the average information content or uncertainty in a random variable. Given a discrete random variable $X$ with a probability mass function $P(X)$, the entropy $H(X)$ can be calculated as follows:

1. Probability Mass Function (PMF): $P(X)$ gives the probability that the random variable $X$ takes on a specific value $x$. For a discrete random variable, $P(X)$ is a function that maps each possible value of $X$ to a probability between 0 and 1, such that the sum of probabilities over all possible values is 1.

2. Entropy Formula: The entropy $H(X)$ is calculated using the PMF:

   $$H(X) = -\sum P(x) \log P(x)$$

   Here's a breakdown of the formula:

   • $\sum$ represents the sum over all possible values $x$ of the random variable $X$.
   • $P(x)$ is the probability of $X$ taking the value $x$.
   • $\log$ is the base-2 logarithm, which is commonly used in information theory. If you prefer to use the natural logarithm (base $e$), you can do so, but you'll need to adjust the base of the logarithm accordingly.

3. Example: Suppose $X$ is a discrete random variable with the following PMF:

   $$P(X) = \begin{cases} 0.2 & \text{if } X = 1 \\ 0.3 & \text{if } X = 2 \\ 0.5 & \text{if } X = 3 \end{cases}$$

   Plugging these values into the entropy formula, we get:

   $$H(X) = - (0.2 \log 0.2 + 0.3 \log 0.3 + 0.5 \log 0.5) \approx 1.52$$