How Would You Differentiate A Discrete From A Continuous Ran

How would you differentiate a discrete from a continuous random variable? Provide a specific example to illustrate the difference.

A discrete random variable is a type of variable that takes on a countable number of distinct values, often integers or whole numbers. In contrast, a continuous random variable can take on any value within a range or interval, including fractional or decimal values, and is uncountably infinite. The primary difference lies in their possible values: discrete variables are countable, while continuous variables are measurable over a continuum.

An example of a discrete random variable is the number of cars serviced at an auto shop in a day. The possible values are whole numbers such as 0, 1, 2, 3, and so on—these are countable quantities. Conversely, the service time for each car is a continuous random variable because it can take any value within a range, such as 15.3 minutes, 16.7 minutes, or 19.2 minutes, measured precisely. This variable can have infinitely many possible values within the interval, making it continuous.

Scenario for Using a Random Sampling Method in Industry

In the auto service industry, suppose a manager wants to estimate the average service time per vehicle during a busy Saturday. To acquire representative data, they might use simple random sampling by selecting cars at random from the scheduled appointments throughout the day. This ensures that every car has an equal chance of being selected, minimizing bias and providing a reliable estimate of the average service time.

I would choose simple random sampling in this scenario because it is straightforward, easy to implement, and ensures each vehicle has an equal probability of selection, leading to a representative sample of the entire day's operations. This method reduces sampling bias and helps in accurately modeling the distribution of service times, which is essential for efficient staffing and resource allocation. Although stratified or systematic sampling could also be used, simple random sampling is ideal here because of the randomness and simplicity offered in a busy, unpredictable environment.

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