Question

Identify the following as discrete or continuous random variables: a. Total number of points scored in a football game b. Shelf life of a particular drug c. Height of the ocean's tide at a given location d. Length of a 2 -year-old black bass e. Number of aircraft near-collisions in a year

Short answer

Question: Classify the given scenarios as either discrete or continuous random variables. a) Total number of points scored in a football game b) Shelf life of a particular drug c) Height of the ocean's tide at a given location d) Length of a 2-year-old black bass e) Number of aircraft near-collisions in a year Answer: a) Discrete random variable b) Continuous random variable c) Continuous random variable d) Continuous random variable e) Discrete random variable

Step-by-step solution

Determine the type of random variable for (a): Total number of points scored in a football game

Football games are scored in whole numbers / countable numbers, and it is not possible to score a fraction of a point. Therefore, the total number of points scored in a football game is a discrete random variable.

Determine the type of random variable for (b): Shelf life of a particular drug

The shelf life of a drug can be represented by a range of values, including fractions of time (days, hours, minutes). It can take on an infinite number of values within that range. Therefore, the shelf life of a particular drug is a continuous random variable.

Determine the type of random variable for (c): Height of the ocean's tide at a given location

The height of the ocean's tide at a given location can be represented by a range of values, including decimals or fractions (meters, centimeters, etc.). It can take on an infinite number of values within that range. Therefore, the height of the ocean's tide at a given location is a continuous random variable.

Determine the type of random variable for (d): Length of a 2-year-old black bass

The length of a 2-year-old black bass can be represented by a range of values, including decimals or fractions (centimeters, millimeters, etc.). It can take on an infinite number of values within that range. Therefore, the length of a 2-year-old black bass is a continuous random variable.

Determine the type of random variable for (e): Number of aircraft near-collisions in a year

The number of aircraft near-collisions in a year will be a whole/countable number since it is not possible to have a fraction of a near-collision. Therefore, the number of aircraft near-collisions in a year is a discrete random variable.

Key concepts

Discrete Random Variables

A discrete random variable is one that can take on a countable number of distinct values. These values are usually whole numbers, which makes them easy to list. Consider the example of the total number of points scored in a football game. Points are only counted in whole numbers, making this a discrete random variable. Other examples include the number of aircraft near-collisions in a year. These events cannot happen in fractions, only in whole instances.

Characteristics of discrete random variables include:

• Countable outcomes
• Finite or infinite sequences of outcomes
• Whole numbers as possible values

Understanding discrete random variables helps in scenarios where listing all possible outcomes makes sense and is feasible.

Continuous Random Variables

A continuous random variable can take an infinite number of possible values within a given range. Unlike discrete variables, these can include both whole numbers and fractions. For example, the shelf life of a drug or the height of the ocean's tide can vary widely without discrete steps. These are continuous because there is no limit to the potential numbers they can be within a given interval.

Some key points about continuous random variables include:

• Uncountable outcomes
• Any value within an interval, including decimals and fractions
• Ranges are often associated with measurements

Recognizing continuous random variables is essential when dealing with measurements and scenarios that require precision.

Probability Theory

Probability theory is the branch of mathematics that studies the likelihood of events happening. It is foundational to understanding both discrete and continuous random variables, as it provides tools to calculate the probabilities of various outcomes.

In probability theory:

• Events are defined as outcomes or sets of outcomes
• The probability of any event is a number between 0 and 1
• A thorough understanding helps in determining the distribution, mean, and variance of random variables

Whether dealing with discrete or continuous variables, probability theory allows us to model real-world scenarios mathematically and make predictions based on data.