Question

Identify the random variables in Exercises \(2-11\) as either discrete or continuous. Number of aircraft near-collisions in a year

Short answer

Answer: The random variable representing the number of aircraft near-collisions in a year is discrete.

Step-by-step solution

Understand Discrete and Continuous Random Variables

Discrete random variables are those variables that can only take a countable number of values, while continuous random variables can take any value within a specific range or interval. In general, if a variable can only take on integer values, it may be considered discrete, while if it can take on any decimal (real) value, it's continuous.

Analyze the Random Variable

In this problem, we are analyzing the number of aircraft near-collisions in a year. The value of this random variable can only be whole numbers (countable) since we cannot have a fraction of a near-collision. For example, we can have zero near-collisions, one near-collision, two near-collisions, and so on - but we cannot have 1.5 near-collisions.

Identify the Random Variable as Discrete or Continuous

Based on the analysis above, we can conclude that the random variable representing the number of aircraft near-collisions in a year is a discrete random variable since it can only take whole number (countable) values.

Key concepts

Continuous Random Variables

When diving into the vast world of probability and statistics, it's essential to understand the difference between discrete and continuous random variables. Continuous random variables can take any value within an interval on the number line. This could be ranges of real numbers like temperatures, weight, or distances. Unlike discrete random variables that are countable and often represented by whole numbers, continuous variables are associated with measurements and can include decimals and fractions.

For example, the time it takes for a student to complete a test could be a continuous random variable because it could be any value within a range, such as between 0 and 120 minutes. This concept is crucial because different types of random variables require different statistical approaches and tools for analysis.

Probability and Statistics

Probability and statistics form the backbone of data analysis, allowing us to make sense of the randomness in the world around us. Probability refers to the measure of the likelihood that an event will occur, and it is quantified as a number between 0 and 1, with 1 indicating certainty. In contrast, statistics is the discipline that concerns the collection, analysis, interpretation, presentation, and organization of data.

Understanding Probability Distributions

In probability and statistics, random variables are often analyzed through their probability distributions—functions that show the probable values that the random variables can take and the frequency of these values. A discrete random variable has a probability mass function (PMF), while a continuous random variable has a probability density function (PDF). The concepts of mean, variance, and standard deviation are also vital in describing the distributions of random variables.

Integer Values

In mathematics and statistics, integer values are whole numbers that can be either positive, negative, or zero, but they cannot include fractions or decimals. Integer values are fundamental when dealing with discrete random variables because these variables take on values that are integer quantities. We can think of them as 'countable' outcomes, such as the number of students in a classroom, the number of cars in a parking lot, or, as in our original exercise, the number of aircraft near-collisions in a year.

It's critical to recognize that not all countable variables are naturally integers. For example, considering the number of children in a family could be a fraction if averaging across a population, but when looking at individual cases, the number of children can only be an integer value. This distinction is why, in probability and statistics, understanding the type of variable—discrete or continuous—is key to correctly applying statistical methods and interpreting results.