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Chapter 8: Problem 55

Are all infinite random variables necessarily continuous? Explain.

Short Answer

Expert verified
No, not all infinite random variables are necessarily continuous. It is possible to have an infinite discrete random variable with countable and distinct outcomes, as demonstrated by the example of the number of coin flips required to obtain the first tails when flipping a fair coin.

Step by step solution

01

Discrete vs Continuous Random Variables

Discrete random variables take on a finite or countable number of distinct values (e.g., the number of heads when flipping a coin). Continuous random variables, on the other hand, can take any value within a range or interval (e.g., the height of a person). In this context, an infinite random variable refers to a random variable that can take on an infinite number of values.
02

Example of an Infinite Discrete Random Variable

Consider a random variable X, which represents the number of coin flips required to obtain the first tails when flipping a fair coin. There is no upper limit on the number of flips required, so this random variable has an infinite number of possible outcomes. However, the outcomes are countable (1, 2, 3, ...) and distinct, which makes this an infinite discrete random variable.
03

Example of an Infinite Continuous Random Variable

Consider the random variable Y, representing the time it takes for a bus to arrive at a given stop. The continuous random variable Y can take any value between 0 and positive infinity, and there are an infinite number of possible outcomes within this range. This example demonstrates an infinite continuous random variable.
04

Conclusion

Not all infinite random variables are necessarily continuous. As demonstrated by the example above, it is possible to have an infinite discrete random variable with countable and distinct outcomes. Therefore, the answer to the question is no.

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Most popular questions from this chapter

Following is a sample of the day-byday change, rounded to the nearest 100 points, in the Dow Jones Industrial Average during 10 successive business days around the start of the financial crisis in October \(2008:^{20}\) $$ -100,400,-200,-500,200,-300,-200,900,-100,200 $$ Compute the mean and median of the given sample. Fill in the blank: There were as many days with a change in the Dow above \(\quad\) points as there were with changes below that.

If we define a "poor" household as one whose after-tax income is at least \(1.3\) standard deviations below the mean, what is the household income of a poor family in Switzerland?

Your company issues flight insurance. You charge \(\$ 2\) and in the event of a plane crash, you will pay out \(\$ 1\) million to the victim or his or her family. In 1989, the probability of a plane crashing on a single trip was \(.00000165\). If ten people per flight buy insurance from you, what was your approximate probability of losing money over the course of 100 million flights in \(1989 ?\) HINT [First determine how many crashes there must be for you to lose money.]

In a certain political poll, each person polled has a \(90 \%\) probability of telling his or her real preference. Suppose that 1,000 people are polled and \(51 \%\) say that they prefer candidate Goode, while \(49 \%\) say that they prefer candidate Slick. Find the approximate probability that Goode could do at least this well if, in fact, only \(49 \%\) prefer Goode.

A roulette wheel has the numbers 1 through 36,0 , and \(00 .\) Half of the numbers from 1 through 36 are red, and a bet on red pays even money (that is, if you win, you will get back your \(\$ 1\) plus another \(\$ 1\) ). How much do you expect to win with a \(\$ 1\) bet on red? HINT [See Example 4.]
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