/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Identify the set of possible val... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 6

Identify the set of possible values for each random variable. (Make a reasonable estimate based on experience, where necessary.) a. The number of hearts in a five-card hand drawn from a deck of 52 cards that contains 13 hearts in all. b. The number of pitches made by a starting pitcher in a major league baseball game. c. The number of breakdowns of city buses in a large city in one week. d. The distance a rental car rented on a daily rate is driven each day. e. The amount of rainfall at an airport next month.

Short Answer

Expert verified
Variable a: \{0, 1, 2, 3, 4, 5\}; Variable b: 0-130; Variable c: 0+; Variable d: 0-500 miles; Variable e: 0+ inches.

Step by step solution

01

Identify Possible Values for Variable a

Consider a five-card hand drawn from a 52-card deck, which includes 13 hearts. The possible number of hearts in a hand could range from 0 (if no hearts are drawn) to 5 (if all drawn cards are hearts). Therefore, the set of possible values is \( \{0, 1, 2, 3, 4, 5\} \).
02

Identify Possible Values for Variable b

In a major league baseball game, starting pitchers usually throw between 0 and about 120 pitches. Sometimes, due to poor performance or injury, a pitcher might be pulled earlier, leading to a range of possible values between 0 and 130 pitches. Therefore, the set of possible values is integers \(0\) to \(130\), inclusive.
03

Identify Possible Values for Variable c

The number of breakdowns of city buses over a week is typically a non-negative integer. There could be 0 breakdowns, or potentially more, depending on the size of the city and the number of buses. Thus, the possible values range from \(0\) to some reasonable upper limit based on experience, such as 50 or more \( \{0, 1, 2, ... , 50, ... \} \).
04

Identify Possible Values for Variable d

The distance driven by a rental car in a day could vary significantly. It could be 0 if the car is not used, or could extend to several hundred miles, possibly around \(400\) to \(500\) miles. Thus, the set includes all non-negative numbers up to a reasonable limit \([0, 500]\) depending on typical driving patterns.
05

Identify Possible Values for Variable e

Rainfall can range from 0 inches (in case of a dry month) to several inches if there are heavy rains. Therefore, the possible values are all non-negative real numbers \([0, \, ext{maximum possible rainfall}]\), where maximum possible must be defined based on historical data for the region.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability Distribution
A probability distribution describes how the values of a random variable are distributed. It gives a full picture of the possible outcomes and the likelihood of each. There are different types, mainly for discrete and continuous variables.
  • Discrete Probability Distribution: Here, the variable can take on distinct, separate values. A good example is rolling a die, where the possible outcomes are the integers 1 through 6.
  • Continuous Probability Distribution: Here, the variable can take on any value within a range. An example is measuring distance or time, where values can include decimals like 1.5 or 2.75.
Understanding probability distributions is essential because they form the foundation for statistical inference and decision-making.
Discrete Variable
Discrete variables are those that can take on a finite or countably infinite set of values. They often appear in contexts where counting is involved, such as the number of hearts in a card hand or pitches a baseball pitcher throws.
  • Discrete variables can only take on specific values: Think of them like steps on a ladder 鈥� you can stop only on set rungs, not in between.
  • Common examples include counts of objects or events, such as the number of buses breaking down or the number of new customers in a store.
Working with discrete variables often involves enumerating all possible outcomes and assigning probabilities to them, forming a probability mass function (PMF).
Continuous Variable
Continuous variables differ from discrete ones because they can take on an infinite number of values within a given range. This means there are no 'gaps' in the possible values.
  • Think of continuous variables as measuring: like volume in a glass, which can be filled to any level within its capacity.
  • Common examples include time duration, temperature, or in our context, the distance a rental car travels.
With continuous variables, we often deal with intervals and use probability density functions (PDFs) to describe the distribution. The total area under a PDF curve represents a probability of 1.
Sample Space
The sample space of a random experiment is the set of all possible outcomes or values of the random variable. It provides a comprehensive overview of what can happen.
  • In card games, the sample space might include different possible hands, such as having 0 to 5 hearts in a drawing.
  • For a pitcher's performance, the possible pitches range from 0 to 130.
Understanding the sample space helps in defining the sample events and is crucial when calculating probabilities, as it lists out every possible scenario that the random variable can support.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Five thousand lottery tickets are sold for \(\$ 1\) each. One ticket will win \(\$ 1,000,\) two tickets will win \(\$ 500\) each, and ten tickets will win \(\$ 100\) each. Let \(X\) denote the net gain from the purchase of a randomly selected ticket. a. Construct the probability distribution of \(X\). b. Compute the expected value \(E(X)\) of \(X .\) Interpret its meaning. c. Compute the standard deviation \(\sigma\) of \(X\).

About \(2 \%\) of alumni give money upon receiving a solicitation from the college or university from which they graduated. Find the average number monetary gifts a college can expect from every 2,000 solicitations it sends.

Let \(X\) denote the number of boys in a randomly selected three-child family. Assuming that boys and girls are equally likely, construct the probability distribution of \(X\).

Determine whether or not the random variable \(X\) is a binomial random variable. If so, give the values of \(n\) and \(p\). If not, explain why not. a. \(X\) is the number of black marbles in a sample of 5 marbles drawn randomly and without replacement from a box that contains 25 white marbles and 15 black marbles. b. \(X\) is the number of black marbles in a sample of 5 marbles drawn randomly and with replacement from a box that contains 25 white marbles and 15 black marbles. c. \(X\) is the number of voters in favor of proposed law in a sample 1,200 randomly selected voters drawn from the entire electorate of a country in which \(35 \%\) of the voters favor the law. d. \(X\) is the number of fish of a particular species, among the next ten landed by a commercial fishing boat, that are more than 13 inches in length, when \(17 \%\) of all such fish exceed 13 inches in length. e. \(X\) is the number of coins that match at least one other coin when four coins are tossed at once.

An English-speaking tourist visits a country in which \(30 \%\) of the population speaks English. He needs to ask someone directions. a. Find the probability that the first person he encounters will be able to speak English. b. The tourist sees four local people standing at a bus stop. Find the probability that at least one of them will be able to speak English.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.