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Chapter 5: Problem 1

Explain the meaning of a random variable, a discrete random variable, and a continuous random variable. Give one example each of a discrete random variable and a continuous random variable.

Short Answer

Expert verified
A random variable is a function that assigns a real number to each outcome of a random experiment. A discrete random variable takes discrete or countable values, like the number of students in a class. A continuous random variable takes infinite number of values within a certain interval, such as height, weight, or time.

Step by step solution

01

Definition of a random variable

A random variable can be defined as a function that maps the outcomes of a random experiment to a real number. In simpler words, whenever an experiment is performed and the result is uncertain, the possible outcomes can be represented as numbers. This representation is termed as a Random variable.
02

Definition of a discrete random variable

A discrete random variable is a type of random variable that can take a countable number of values. These values do not necessarily have to be integers, but they must be separable, meaning that there are gaps between them. For instance, the number of items sold in a supermarket, number of students in a class, number of heads when flipping three coins - all are examples of discrete random variables.
03

Definition of a continuous random variable

A continuous random variable, on the other hand, can take an infinite number of values within a certain interval. These values often result from measurements and can take on any value in the interval. Examples include the height of people, weight of a batch of fruit, time elapsed between two events, etc.

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Most popular questions from this chapter

A limousine has eight tires on it. A fleet of such limos was fit with a batch of tires that mistakenly passed quality testing. The following table lists the probability distribution of the number of defective tires on this fleet of limos where \(x\) represents the number of defective tires on a limo and \(P(x)\) is the corresponding probability. $$ \begin{array}{l|ccccccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline P(x) & .0454 & .1723 & .2838 & .2669 & .1569 & .0585 & .0139 & .0015 & .0008 \\ \hline \end{array} $$ Calculate the mean and standard deviation of this probability distribution. Give a brief interpretation of the values of the mean and standard deviation.

A really bad carton of 18 eggs contains 7 spoiled eggs. An unsuspecting chef picks 4 eggs at random for his "Mega-Omelet Surprise." Find the probability that the number of unspoiled eggs among the 4 selected is a. exactly \(\overline{4}\) b. 2 or fewer c. more than 1

Let \(x\) be the number of houses sold per month by a real estate agent. The following table lists the probability distribution of \(x\). $$ \begin{array}{l|cccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(x) & .08 & .12 & .32 & .28 & .12 & .08 \\ \hline \end{array} $$ Calculate the mean and standard deviation of this probability distribution and give a brief interpretation of the value of the mean.

During the 2014 NFL regular season, kickers converted \(88 \%\) of the field goals attempted. Assume that this percentage is true for all kickers in the upcoming NFL season. Find the probability that a randomly selected kicker who will try 4 field goal attempts in a game will a. convert all 4 field goal attempts b. miss all 4 field goal attempts

Five percent of all cars manufactured at a large auto company are lemons. Suppose two cars are selected at random from the production line of this company. Let \(x\) denote the number of lemons in this sample. Write the probability distribution of \(x\). Draw a tree diagram for this problem.
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