A random variable is a numerical description of the outcomes of a random phenomenon. Think of it as a variable that can take different values, determined by the randomness of an experiment or event.
There are two types of random variables: discrete and continuous. Our focus here is on discrete random variables, which take on a finite or countably infinite set of values. In the exercise, the random variable \(X\) can take on any of the values \(1, 2, 3, \ldots, k\).
- Each of these values corresponds to an event whose probability is given by the PMF.
- Understanding how a random variable behaves helps us compute its MGF, which is derived from these probabilities.
The discrete random variable's behavior is the basis for creating various probability models and predicting possible outcomes, allowing for deeper analysis in the field of probability and statistics.