Question

Identify the random variables in Exercises \(2-11\) as either discrete or continuous. Total number of points scored in a football game

Short answer

Answer: The random variable is discrete because it only takes whole-number values and not decimal values as it is countable.

Step-by-step solution

Understanding Discrete and Continuous Random Variables

Discrete random variables are variables that take on countable (usually whole number) values, like rolling a die (you only get whole numbers as results) or counting something. Continuous random variables, on the other hand, can take on any value within a specific range or interval, like measuring the time it takes for an event to occur or the height of a person (which can have decimal values).

Identify the given random variable

The random variable given in this exercise is the "Total number of points scored in a football game." In a football game, the points are scored as whole numbers (like a touchdown equals 6 points, a field goal equals 3 points, etc.). There is no possibility to score decimal values (for example, 2.5 points).

Classify the random variable

Based on the previous understanding of the given random variable, it is clear that the "Total number of points scored in a football game" is a discrete random variable because it only takes whole-number values and not decimal values as it is countable.

Key concepts

Continuous Random Variables

In probability and statistics, understanding the distinction between discrete and continuous random variables is fundamental. A continuous random variable, unlike its discrete counterpart, can take on any value within a specified range. This includes not only whole numbers but also decimals and even irrational numbers.

Examples of continuous random variables are abundant in real life. Consider the measurement of time, temperature, or the weight of an object. These can be quantified with extreme precision, and as such, they require the use of continuous random variables for accurate representation. If you were to measure the length of a leaf, the possible outcomes are infinite within a certain range – it could be 5.345 cm, 5.346 cm, and so on.

To handle the calculation of probabilities with continuous random variables, we utilize probability density functions (PDF) which differ from the probability mass functions used for discrete variables. PDFs help in calculating the likelihood that a continuous random variable falls within a certain range, rather than at specific points, since identifying the probability at a precise point for a continuous variable is practically zero due to the infinite possibilities within the range.

Probability and Statistics

Probability and statistics are intertwined disciplines that involve the study of uncertainty and data analysis. Probability is the branch of mathematics that deals with calculating the likelihood of a given event’s occurrence, which could be as simple as tossing a coin or as complex as predicting weather patterns.

In the context of our exercise, statistics would come into play when attempting to analyze the scores of numerous football games. By compiling data on scores, statisticians can determine averages, variances, and other descriptive measures. Moreover, this field helps in making inferences about a larger population based on sample data, such as predicting the likely range of scores in future games. Statistical methods enable us to understand and interpret data, foresee trends, and make decisions based on the inherent randomness of the events being studied.

Quantitative Variable Classification

Quantitative variables represent data that can be quantified and are critical in statistical analysis. The classification of these variables greatly assists in deciding upon the appropriate statistical methods for data analysis. There are two primary classifications of quantitative variables: discrete and continuous.

As highlighted in the football game scoring example, discrete random variables take on set numeric values and are countable. These include things like the number of students in a class, the number of cars in a parking lot, or the total number of points scored in a football game. Here, decimal values do not make sense as you can’t have half a student or half a car.

In contrast, continuous variables are those that can take on an infinite number of values within a specific range. Height, weight, and temperature are examples where precise measurements are not only possible but necessary. The choice of the variable type affects how researchers collect data and how they analyze it. For instance, different types of charts and graphs are used to visualize these different types of variables, with histograms often used for discrete data and line graphs for continuous data.