Final answer:
A random variable is a numerical outcome of a probability experiment, which can be discrete (countable) or continuous (measurable). Examples include modeling the direction of car travel as a discrete variable and the time it takes for cars to travel a distance as a continuous variable.
Step-by-step explanation:
Understanding Random Variables and Probability Models
A random variable is a numerical outcome of a probability experiment, where its value is not predetermined but instead arises from the randomness inherent in the experiment itself. We can distinguish between two main types of random variables: discrete random variables and continuous random variables.
Discrete random variables are those variables whose values are countable, such as the number of cars traveling right, or the color of each car. These can be represented using a discrete probability distribution, where the probability of each possible value is between zero and one, and the total sum of all probabilities equals one.
Continuous random variables, on the other hand, are not countable but measurable, such as the time it takes for a car to go a specific distance. These can be modeled using continuous probability distributions, which can take on any value within an interval on the number line. Examples of continuous random variables include height, weight, temperature, and time.
For instance, to model the direction each car is traveling, one could use a discrete random variable with two outcomes: right or left, assigning probabilities based on observed frequencies. The total number of cars traveling right out of 100 cars could be modeled as a binomial distribution, where each car has a fixed probability of turning right. When considering the color of each car, one could use a categorical random variable if there are a limited number of colors, assigning probabilities for each color. Lastly, the time it takes for each car to go the length of the block would be modeled as a continuous random variable, which could follow normal distribution if the times are symmetrically distributed around a mean time, or another appropriate distribution depending on the data.