Final answer:
Discrete random variables have countable outcomes, while continuous random variables have an infinite number of possible values within a range. Discrete random variables' probabilities must total one and fall between zero and one, whereas continuous random variables are defined over a range due to their uncountable nature.
Step-by-step explanation:
Discrete and continuous random variables are two fundamental concepts in probability and statistics. A discrete random variable has a countable number of possible values. For instance, the number of heads you get when flipping a coin multiple times or the number of red balls drawn from a bag are discrete because you can count the outcomes. Importantly, the probability distribution function (PDF) for a discrete variable must satisfy two conditions: each probability must be between zero and one, inclusive, and the sum of all probabilities must be one.
On the other hand, a continuous random variable takes on an uncountable number of possible values, representing measurements. Examples include the temperature on a given day or the height of students, which can have many possible values within a range. Probabilities for continuous variables are not calculated for specific values but for a range of values because it is essentially impossible for the variable to take on any one precise value due to the infinite possibilities within real numbers.
Understanding how to define these variables and their respective probability distributions is crucial for solving real-world statistical problems and interpreting data correctly