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Chapter 3: Problem 5

If the sample space \(\delta\) is an infinite set, does this necessarily imply that any rv \(X\) defined from \(s\) will have an infinite set of possible values? If yes, say why. If no, give an example.

Short Answer

Expert verified
No, a random variable can have finite values even if the sample space is infinite.

Step by step solution

01

Understanding the Problem

We need to determine if a random variable (rv) \(X\) defined from an infinite sample space \(\delta\) will always have an infinite set of possible values. Essentially, we're asked to consider whether the nature of the sample space directly influences the set of possible values for \(X\).
02

Analyze Sample Space and Random Variable

Recognize that the sample space \(\delta\) refers to all possible outcomes of an experiment. A random variable \(X\) is a function that assigns numerical values to outcomes in \(\delta\). The key question is whether infinite outcomes in \(\delta\) lead to infinite values of \(X\).
03

Examine Potential Counterexamples

Consider counterexamples where the sample space \(\delta\) is infinite, but \(X\) only takes a limited number of values. For instance, let's define \(\delta = \{1, 2, 3, \ldots\}\) an infinite set, and define \(X\) such that \(X(\delta) = 0\) for all values in \(\delta\).
04

Evaluate the Counterexample

In the example given, even though \(\delta\) is infinite, \(X\) takes only one value, 0. This demonstrates that \(X\) does not necessarily have an infinite set of possible values even if \(\delta\) is infinite.
05

Conclusion

The existence of a counterexample shows that an infinite sample space does not necessarily mean that a random variable defined on it has an infinite set of values.

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Key Concepts

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

Infinite Sample Space
When discussing an infinite sample space, we are referring to a set that contains an unlimited number of potential outcomes. In probability theory, this means there could be infinitely many results from an experiment. Think of flipping a fair coin repeatedly鈥攖he sequence of heads and tails can continue indefinitely, creating an infinite sample space.

However, the vastness of a sample space does not necessitate that any corresponding random variable (rv) will also have infinite values. A random variable is essentially a rule or formula that assigns numerical values to these outcomes. For instance, imagine a scenario where you roll a die continuously without stopping. The potential results (like the total count of rolls) are infinite. Yet, if your random variable is defined simply as whether the count is odd or even, the outcomes are finite (just two possible values).
  • Infinite sample spaces can be continuous (like measuring a person's height) or discrete (like counting dice rolls).
  • The sample space provides the foundational set of outcomes on which random variables operate.
  • Understanding that infinite doesn't always mean complex or numerous for a random variable is crucial.
Discrete Random Variable
A discrete random variable is a type of random variable that can take on a countable number of distinct values. This means its potential outcomes can be listed, even if they are infinitely numerous, like the set of natural numbers. Discrete random variables are often the focus when dealing with probability questions involving counting or categorization.

For example, consider a discrete random variable representing the number of times a red ball is drawn from a bag after several tries. This rv could be {0, 1, 2, 3, ...} and so on. Even though it might seem like there are endless possibilities, each outcome is individually countable.
  • Discrete random variables can measure things like counts or categories.
  • They depend on a sample space, but not necessarily an infinite one.
  • Utilizing discrete random variables can simplify complex systems into manageable sets of outcomes.
Probability Theory
Probability theory is the mathematical framework we use to study randomness and uncertainty. It helps us quantify how likely an event is to occur within a given sample space. This theory is pivotal for understanding how infinite sample spaces and random variables interact.

Within probability theory, every outcome of an experiment has associated probabilities that sum up to one. When dealing with infinite sample spaces, assigning probabilities can sometimes involve complex mathematical models, but often we still rely on simplifying assumptions. Probability theory helps establish the relationship between a sample space and the random variables defined on it.
  • Probability theory includes the concepts of sample spaces, random variables, and probability distributions.
  • It provides tools to calculate and predict outcomes in random systems.
  • By applying probability theory, you can assess scenarios with infinite possibilities in a structured way.

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Most popular questions from this chapter

Suppose that the number of plants of a particular type found in a rectangular region (called a quadrat by ecologists) in a certain geographic area is an rv \(X\) with pmf $$ p(x)=\left\\{\begin{array}{cc} c / x^{3} & x=1,2,3, \ldots \\ 0 & \text { otherwise } \end{array}\right. $$ Is \(E(X)\) finite? Justify your answer (this is another distribution that statisticians would call heavy-tailed).

Two fair six-sided dice are tossed independently. Let \(M=\) the maximum of the two tosses [thus \(M(1,5)=5, M(3,3)=3\), etc.]. a. What is the pmf of \(M\) ? [Hint: First determine \(p(1)\), then \(p(2)\), and so on.] b. Determine the cdf of \(M\) and graph it.

Define a function \(p(x ; \lambda, \mu)\) by $$ \begin{aligned} &p(x ; \lambda, \mu) \\ &=\left\\{\begin{array}{cc} \frac{1}{2} e^{-\lambda} \frac{\lambda^{x}}{x !}+\frac{1}{2} e^{-\mu} \frac{\mu^{x}}{x !} & x=0,1,2, \ldots \\ 0 & \text { otherwise } \end{array}\right. \end{aligned} $$ a. Show that \(p(x, \lambda, \mu)\) satisfies the two conditions necessary for specifying a pmf. [Note: If a firm employs two typists, one of whom makes typographical errors at the rate of 驴 per page and the other at rate \(\mu\) per page and they each do half the firm's typing, then \(p(x ; \lambda, \mu)\) is the pmf of \(X=\) the number of errors on a randomly chosen page.] b. If the first typist (rate \(\lambda\) ) types \(60 \%\) of all pages, what is the penf of \(X\) of part (a)? c. What is \(E(X)\) for \(p(x, \lambda, \mu)\) given by the displayed expression? d. What is \(\sigma^{2}\) for \(p(x ; \lambda, \mu)\) given by that expression?

Suppose that you read through this year's issues of the New York Times and record each number that appears in a news article-the income of a CEO, the number of cases of wine produced by a winery, the total charitable contribution of a politician during the previous tax year, the age of a celebrity, and so on. Now focus on the leading digit of each number, which could be \(1,2, \ldots, 8\), or 9 . Your first thought might be that the leading digit \(X\) of a randomly selected number would be equally likely to be one of the nine possibilities (a discrete uniform distribution). However, much empirical evidence as well as some theoretical arguments suggest an alternative probability distribution called Benford's law: $$ \begin{aligned} p(x) &=P(1 \text { st digit is } x)=\log _{10}\left(\frac{x+1}{x}\right), \\ x &=1,2, \ldots, 9 \end{aligned} $$ a. Without computing individual probabilities from this formula, show that it specifies a legitimate pmf. b. Now compute the individual probabilities and compare to the corresponding discrete uniform distribution. c. Obtain the cdf of \(X\). d. Using the odf, what is the probability that the leading digit is at most 3 ? At least 5 ? [Note: Benford's law is the basis for some auditing procedures used to detect fraud in financial reporting-for example, by the Internal Revenue Service.]

An instructor who taught two sections of statistics last term, the first with 20 students and the second with 30 , decided to assign a term project. After all projects had been tumed in, the instructor randomly ordered them before grading. Consider the first 15 graded projects. a. What is the probability that exactly 10 of these are from the second section? b. What is the probability that at least 10 of these are from the second section? c. What is the probability that at least 10 of these are from the same section? d. What are the mean value and standard deviation of the number among these 15 that are from the second section? e. What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section?
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