Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The number of boys in a four-child family

Step-by-Step Solution

Answer

The random variable \(X\) can assume the values in the set \(\{0, 1, 2, 3, 4\}\), representing the number of boys in a four-child family. \(X\) is classified as a finite discrete random variable, as it has a finite number of outcomes and can only assume whole number values.

Step 1: Determine the Range of Values

To find the range of values that \(X\) can take, we consider the possible outcomes for the gender of each child in a family with four children. For each child, there are two options: boy (B) or girl (G). So, the outcomes for a four-child family are: 1. Four boys: (B, B, B, B) 2. Three boys, one girl: (B, B, B, G), (B, B, G, B), (B, G, B, B), (G, B, B, B) 3. Two boys, two girls: (B, B, G, G), (B, G, B, G), (B, G, G, B), (G, B, B, G), (G, B, G, B), (G, G, B, B) 4. One boy, three girls: (B, G, G, G), (G, B, G, G), (G, G, B, G), (G, G, G, B) 5. Four girls: (G, G, G, G) From these outcomes, we can see that the number of boys, \(X\), can assume the following values: \[X = \{0, 1, 2, 3, 4\}\]

Step 2: Classify the Random Variable

The random variable \(X\) represents the number of boys in a family with four children and can take on the values in the set \(\{0, 1, 2, 3, 4\}\). Since \(X\) has a finite number of outcomes, it is classified as a finite discrete random variable. The variable cannot be infinite because the number of children in the family is limited to four, and it is not continuous because it can only assume whole number values.

Key Concepts

Probability Distribution

Understanding the concept of a probability distribution is crucial when studying random variables. In essence, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes for an experiment.

For a discrete random variable like the number of boys in a four-child family, a probability distribution would list all the possible values that the variable can assume paired with the likelihood of each value occurring. It assists in showing how probabilities are spread over the values of the random variable. The distribution is usually represented in the form of a table, graph, or formula.

It's important to note that the probabilities of all possible outcomes must add up to 1, as this represents the certainty that one of the possible outcomes will occur. In the example provided, the distribution would show the probabilities of having 0, 1, 2, 3, or 4 boys in a four-child family.

Finite Discrete Random Variable

The term 'finite discrete random variable' refers to a random variable that possesses a countable number of distinct outcomes. 'Finite' denotes that there's a limited number of potential outcomes, while 'discrete' implies that these outcomes are separate and distinct—think of them as countable items, like rolling a die and getting a whole number from 1 to 6.

Applying this to our four-child family example, the random variable representing the number of boys is finite because there can only be a maximum of four boys, corresponding to the number of children in the family. It is discrete since you can have only an integer number of boys—no family can have 2.5 boys, for example. This classification helps inform how to handle the variable in a statistical or probability context, particularly when it comes to calculating probabilities and other related measures.

Random Variable Classification

Classification aids in better understanding and applying statistical methods to random variables. Besides being discrete or continuous, random variables can also be classified based on their range—finite or infinite.

If a random variable can only take on values within a specified finite set, like the number of boys in a family of four, it is termed a 'finite random variable.' Alternatively, an 'infinite discrete random variable' would allow for an infinite sequence of values, such as the number of customers arriving at a store in a day, which theoretically could be any integer. Finally, 'continuous random variables' do not have distinct separate values but can take on any value within a certain range, such as the weight of a randomly selected individual.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints. In the context of probability and random variables, combinatorics helps to calculate the number of possible outcomes of an event.

For example, when determining the number of ways boys and girls can be born in a four-child family, combinatorics allows us to enumerate the various combinations of boys (B) and girls (G), from all boys to all girls. This branch of mathematics is pivotal in understanding the foundation of probabilities, especially when dealing with discrete random variables. In real-world applications, mastering combinatorics can help solve problems ranging from predicting genetic variations to optimizing computer network configurations.