Question

Identify the random variables in Exercises \(2-11\) as either discrete or continuous. Increase in length of life attained by a cancer patient as a result of surgery

Short answer

Answer: The increase in length of life attained by a cancer patient as a result of surgery is a continuous random variable.

Step-by-step solution

Identifying the nature of the variable

For the increase in the length of life attained by a cancer patient as a result of surgery, we need to think about the possible values this variable may have. Length of life can be measured in various units like days, months, or even fractions of a second, so it can take on any value within an interval. Therefore, this variable would be considered continuous since it doesn't have a finite or countable number of possible values.

Final conclusion

The increase in length of life attained by a cancer patient as a result of surgery can be considered a continuous random variable since it can take on any value within a specified range.

Key concepts

Discrete Random Variable

A discrete random variable is a type of random variable that can take on a countable number of distinct values. This countability means that we can enumerate the possible outcomes, whether it's a finite number or an infinite sequence that can be listed like natural numbers.

For instance, consider a six-sided die. When rolling the die, the number that comes up is a discrete random variable because there are only six possible outcomes: 1, 2, 3, 4, 5, or 6. Each of these outcomes is distinct and has a probability associated with it. Similarly, the number of students in a classroom, the number of cars in a parking lot, or the number of hurricanes in a hurricane season are all examples of discrete random variables. They are often represented in a probability distribution table or a bar graph, reflecting the discrete nature of their possible values.

Random Variables in Statistics

In the realm of statistics, a random variable is considered a variable whose values depend on the outcomes of a random phenomenon. It's like a placeholder that captures the result of a stochastic, or random, process. Random variables are broadly classified into two types: discrete and continuous.

The main distinction lies in the types of outcomes they can take on. A discrete random variable has specific, countable outcomes, as mentioned before. On the other hand, a continuous random variable, such as the one in the given exercise (increase in length of life), can assume any value in a continuous range. Being adept at distinguishing between these types is crucial for students and practitioners in the field because it determines the methods used for calculating probabilities, statistical inferences, and interpreting results.

Identifying Random Variables

To identify whether a random variable is discrete or continuous, consider the nature of what's being measured and how precise the measurement is. If it involves counting and the results can be listed, it's discrete. For measurements that are inherently quantifiable only by intervals, the random variable is continuous.

For example, the number of texts a person receives per day is discrete – there's no such thing as half a text. However, the amount of time spent on a phone call, which can be measured down to fractions of a second, is a continuous random variable. Identifying random variables correctly is essential because it determines the type of probability distribution – discrete or continuous – that is relevant for the variable. Students, while working on problems, should ask whether the variable's possible values are countable or uncountably infinite. This understanding lays the foundation for proper analysis and application of statistical techniques.