The basic characteristic of a discrete random variable is the existence of a discontinuity between two adjacent values that the random variable can take.

The values that a random variable can take are finite or countably infinite. In addition, the values of a random variable are not limited to the set of natural numbers and 0, but can take on isolated values from the set of real numbers.

Discontinuous random variables occur in many situations:

  • the number of students in a class,
  • the result of rolling a dice,
  • the number of rainy days in March, etc.

In simpler cases, such as rolling a dice, the possible values of a random variable are easily identified by simple counting. However, in the general case, this may not be practical, because the set of possible values of a random variable is very large. Therefore, such a set is described by a mathematical function called the distribution function.

A discrete random variable is most often represented graphically as a bar chart or probability histogram. The values of the random variable are on the X-axis, and the probabilities are on the Y-axis.

The cumulative distribution function sums the probabilities of events occurring.

Example: Cumulative Probability Distribution

What is the probability that a number less than or equal to 4 is obtained by rolling a die?

Answer: The random variable can take on the values 1, 2, 3, 4, 5, and 6. The probability of each of these events is the same and is p=1/6. In our example, by applying the rule of adding probabilities, we get that the probability of rolling a die with a number less than 5 is:

P(X <= 4) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1/6 + 1/6 + 1/6 + 1/6 = 4/6 = 2/3, or 66.67%