Problem 8 - 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 defective watches in a sample of eight watches

Step-by-Step Solution

Answer

The range of values for the random variable \(X\) is between 0 and 8 inclusive, as it represents the number of defective watches in a sample of eight watches. It can assume any integer value in this range. The random variable \(X\) is classified as a finite discrete random variable because it takes on a finite number of distinct values.

Step 1: Determine the range

X = number of defective watches in a sample of 8. The possible values are \( \{0, 1, 2, 3, 4, 5, 6, 7, 8\} \).

Step 2: Classify

Since X takes a finite number of integer values (0 through 8), it is a finite discrete random variable.

Key Concepts

Finite Discrete Random Variable

When we encounter the term finite discrete random variable, it indicates a specific type of variable used in statistics and probability theory. A random variable, to put it simply, is a variable whose values are determined by the outcomes of a random phenomenon. What characterizes a variable as discrete is that it can only take on specific, separate values. These values can be counted, often because they represent occurrences, such as the number of cars passing by a checkpoint, or, as in our original problem, the number of defective watches in a sample.

Furthermore, the 'finite' part of finite discrete random variable implies that there is a limited number of possible outcomes. In the given exercise, for instance, the number of defective watches in a sample of eight watches can only range from 0 to 8, which is a finite set of nine possible outcomes. Each of these outcomes is discrete and distinct, which makes the random variable finite and discrete.

To identify such a variable, we typically ask if the outcomes are countable and if there is a clear upper limit to the possible values. If the answer to both is yes, as with our example of the defective watches (0 to 8), we have a finite discrete random variable.

Range of a Random Variable

The range of a random variable is a fundamental concept that defines all the possible values it can assume. In the textbook problem, the task was to determine the range for the random variable representing defective watches in a sample. This random variable, denoted by \(X\), has a range from the smallest possible value, which in this context is 0 (if there are no defective watches), to the largest possible value, which is 8 (if every watch in the sample is defective).

It is essential to note that the range consists only of values that can actually occur. Therefore, the exercise's intimation to establish a range refers to listing out all integers from 0 to 8. Each number corresponds to a potential outcome, emphasizing that the variable cannot assume any value outside this defined range. The range is not to be confused with an interval on the number line; it is a discrete set of outcomes. In applied scenarios, understanding the range helps in constructing probability distributions and forecasts based on the random variable.

Probability Distribution

Delving into the notion of probability distribution takes us a step further in grasping the behavior of random variables. It essentially describes how probabilities are allocated across the possible values of a random variable. For our finite discrete random variable \(X\), the probability distribution would specify the likelihood of \(X\) being equal to each number in its range, from 0 to 8 in our example. The probabilities assigned to each possible outcome must satisfy two criteria: each probability must be between 0 and 1, and all probabilities must sum up to 1.

For instance, if \(X\) referred to the number of heads when flipping a coin three times, the probability distribution might show that there's a 12.5% chance of getting no heads, a 37.5% chance of getting one head, and so forth. These figures help to predict outcomes and understand the likelihood of various scenarios. In constructing such distributions for finite discrete random variables, formulas from combinatorics and principles of statistical analysis are often used to ensure that all probabilities are accurately allocated.