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Chapter 8: Problem 12

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

Short Answer

Expert verified
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 by step solution

01

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: \[X = \{0, 1, 2, 3, 4\}\]
02

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.

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