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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 assigns numerical values to each outcome of a random process. A discrete random variable has countable outcomes, like the number of heads when flipping three coins. A continuous random variable has an infinite array of possible outcomes within an interval, like commute times between two points.

Step by step solution

01

Define Random Variable

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In other words, it's a way to associate a unique numerical value with each outcome of some random process.
02

Define Discrete Random Variable

A discrete random variable is a random variable for which the outcomes are countable. This includes any set of values that you can list out, even if the list goes on indefinitely.
03

Example of a Discrete Random Variable

An example of a discrete random variable would be the number of heads when flipping three coins. The possible outcomes are 0, 1, 2, or 3.
04

Define Continuous Random Variable

A continuous random variable is a random variable with an entire interval of numbers for its potential outcomes. These variables typically represent measurements, and their values are not countable as there are infinitely many possibilities.
05

Example of a Continuous Random Variable

An example of a continuous random variable could be the time it takes for a commuter to reach their office. The exact time could be any real number between a certain time range, say between 0 and 2 hours.

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Most popular questions from this chapter

Although Microsoft Windows is the primary operating system for desktop and laptop PC computers, Microsoft's Windows Phone operating system is installed in only \(1.6 \%\) of smartphones (www. latimes.com/business/la-fi-google- mobile-20110817,0,6230477.story). a. Assuming that \(1.6 \%\) of all current smartphones have Microsoft's Windows Phone operating system, using the binomial formula, find the probability that the number of smartphones in a sample of 80 that have Microsoft's Windows Phone operating system is i. exactly 2 ii. exactly 4 b. Suppose that \(5 \%\) of all current smartphones have Microsoft's Windows Phone operating system. Use the binomial probabilities table (Table I of Appendix C) or technology to find the probability that in a random sample of 20 smartphones, the number of smartphones that have Microsoft's Windows Phone operating system is i. at most 2 ii. 2 to 3 iii. at least 2

One of the four gas stations located at an intersection of two major roads is a Texaco station. Suppose the next six cars that stop at any of these four gas stations make their selections randomly and independently. Let \(x\) be the number of cars in these six that stop at the Texaco station. Is \(x\) a discrete or a continuous random variable? Explain. What are the possible values that \(x\) can assume?

Although Borok's Electronics Company has no openings, it still receives an average of \(3.2\) unsolicited applications per week from people seeking jobs. a. Using the Poisson formula, find the probability that this company will receive no applications next week. b. Let \(x\) denote the number of applications this company will receive during a given week. Using the Poisson probabilities table from Appendix \(\mathrm{C}\), write the probability distribution table of \(x\). c. Find the mean, variance, and standard deviation of the probability distribution developed in part \(\mathrm{b}\).

According to the Alzheimer's Association (www.alz.org/documents_custom/2011_Facts_Figures Fact_Sheet.pdf), \(3.7 \%\) of Americans with Alzheimer's disease were younger than the age of 65 years in 2011 (which means that they were diagnosed with early onset of Alzheimer's). Suppose that currently \(3.7 \%\) of Americans with Alzheimer's disease are younger than the age of 65 years. Suppose that two Americans with Alzheimer's disease are selected at random. Let \(x\) denote the number in this sample of two Americans with Alzheimer's disease who are younger than the age of 65 years. Construct the probability distribution table of \(x\). Draw a tree diagram for this problem.

An average of \(.8\) accident occur per day in a particular large city. a. Find the probability that no accident will occur in this city on a given day. b. Let \(x\) denote the number of accidents that will occur in this city on a given day. Write the probability distribution of \(x\). c. Find the mean, variance, and standard deviation of the probability distribution developed in part b.
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