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Chapter 5: Problem 1

Define and give three examples of a random variable.

Short Answer

Expert verified
A random variable assigns values to outcomes in a sample space. Examples include a die roll, time taken for a task, and a coin flip result.

Step by step solution

01

Defining a Random Variable

A random variable is a numerical description of the outcome of a statistical experiment. It is a function that assigns a real number to each outcome in a sample space. Random variables can be discrete or continuous, depending on the nature of the values they can take.
02

Example 1 - Discrete Random Variable

Consider a six-sided die. The random variable X, representing the outcome of a die roll, can take on any one of the set of values {1, 2, 3, 4, 5, 6}, each associated with a probability of 1/6. This is an example of a discrete random variable, since it takes on distinct, separate values.
03

Example 2 - Continuous Random Variable

Imagine measuring the time it takes for a student to complete a math test, which can be any positive real number. Define Y as the time in minutes. Since Y can take on an infinite number of values within a given range, it is an example of a continuous random variable.
04

Example 3 - Binary Random Variable

Let's examine a coin flip. Define Z as a binary random variable representing 'heads' (0) or 'tails' (1). The possible values of Z are discrete, but limited to two, hence it's also discrete. Binary random variables are a specific type of discrete variable with only two possible outcomes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Random Variable
A discrete random variable is like stepping on stones across a river. Each stone represents a distinct, countable value the variable might take. Imagine rolling a fair six-sided die. The number you roll, represented as a variable \(X\), can be 1, 2, 3, 4, 5, or 6. That's a finite set of possible results.

In summary:
  • Discrete random variables have specific, individual outcomes.
  • They are countable, like the numbers of a die roll or the number of cars passing a point in an hour.
  • Each value is associated with a distinct probability.
Understanding this concept helps in calculating probabilities for situations involving limited distinct outcomes.
Continuous Random Variable
Continuous random variables are like a smooth mountain slope, offering infinite possibilities. They allow variables to assume any value within a certain range. Consider measuring the exact time a student finishes a test, which could be \(5.1\), \(5.12\), or any fractional value of minutes. The variable \(Y\) has endless possibilities along the time scale.

Here are some key points:
  • Continuous random variables can take on any value within a range.
  • They are not countable in the same way discrete variables are, since they cover an infinite span of values.
  • Sometimes, probabilities for continuous variables are considered over an interval, rather than at specific points.
This forms the basis for modeling real-world phenomena that change fluidly, like temperature variations or time measurements.
Binary Random Variable
Binary random variables simplify life into two choices, much like a light switch being on or off. These variables have only two possible outcomes, most commonly labeled as 0 and 1. Flipping a coin captures this perfectly. Let \(Z\) be the outcome, with a heads being 0 and tails being 1. The binary nature means there are only two potential results, where each has its own probability.

Consider these aspects:
  • Binary variables fall under the discrete category, given their finite outcome set.
  • They are useful in modeling situations where only two exclusive possibilities exist, like true/false questions.
  • Binary outcomes are crucial for simple decision-making processes and for algorithms in computing.
Grasping this concept is fundamental for analyzing basic probabilities and is foundational for more complex statistical models.

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Most popular questions from this chapter

Automobiles A survey shows the probability of the number of automobiles that families in a certain housing plan own. Find the mean, variance, and standard deviation for the distribution. $$ \begin{array}{|c|ccccc|}\hline X & {1} & {2} & {3} & {4} & {5} \\ \hline P(X) & {0.27} & {0.46} & {0.21} & {0.05} & {0.01} \\ \hline\end{array} $$

Job Applications Ten people apply for a job at Computer Warehouse. Five are college graduates and five are not. If the manager selects 3 applicants at random, find the probability that all 3 are college graduates.

Survey of High School Seniors Of graduating high school seniors, 14% said that their generation will be remembered for their social concerns. If 7 graduating seniors are selected at random, find the probability that either 2 or 3 will agree with that statement.

Another type of problem that can be solved uses what is called the negative binomial distribution, which is a generalization of the binomial distribution. In this case, it tells the average number of trials needed to get k successes of a binomial experiment. The formula is $$ \mu=\frac{k}{p} $$ $$ \begin{array}{c}{\text { where } k=\text { the number of successes }} \\\ {p=\text { the probability of a success }}\end{array} $$ Use this formula for Exercises 27鈥�30. Drawing Cards A card is randomly drawn from a deck of cards and then replaced. The process continues until 3 clubs are obtained. Find the average number of trials needed to get 3 clubs.

Computer Games The probability that a child plays one computer game is one-half as likely as that of playing two computer games. The probability of playing three games is twice as likely as that of playing two games, and the probability of playing four games is the average of the other three. Let X be the number of computer games played. Construct the probability distribution for this random variable and draw the graph.
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