Final answer:
A variable that can assume any value on a given interval is known as a continuous random variable, which is measured rather than counted. Probabilities for these variables are calculated based on ranges due to the infinite number of possible values within any interval.
Step-by-step explanation:
When the variable can theoretically assume any value on a given interval, it is referred to as a continuous random variable. Unlike discrete random variables which are countable and obtained by counting, continuous variables are measured and uncountable. An example is the temperature of a day, which can take any value within a range and is measured using a thermometer. Height is another example, as it can vary to a great degree of precision and is measured using a ruler.
Continuous random variables are essential in many fields, and finding probabilities involves specifying ranges since the exact probability of assuming any particular value, due to the infinite possibilities within an interval, is technically zero. Instead, we discuss the probability of the variable falling within a certain range.
Mathematically, a uniform distribution is an example where the random variable has equally likely outcomes within the interval a < x < b. This concept is crucial in understanding how probabilities are assigned and calculated for continuous variables.