/*! 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 7 For each random variable defined... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 7

For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete. a. \(X=\) the number of unbroken eggs in a randomly chosen standard egg carton b. \(Y=\) the number of students on a class list for a particular course who are absent on the first day of classes c. \(U=\) the number of times a duffer has to swing at a golf ball before hitting it d. \(X=\) the length of a randomly selected rattlesnake e. \(Z=\) the amount of royalties earned from the sale of a first edition of 10,000 textbooks f. \(Y=\) the \(\mathrm{pH}\) of a randomly chosen soil sample g. \(X=\) the tension (psi) at which a randomly selected tennis racket has been strung h. \(X=\) the total number of coin tosses required for three individuals to obtain a match (HHH or TTT)

Short Answer

Expert verified
a. Discrete; b. Discrete; c. Discrete; d. Continuous; e. Continuous; f. Continuous; g. Continuous; h. Discrete.

Step by step solution

01

Analyze the Random Variable X (number of unbroken eggs)

The variable \( X \) represents the number of unbroken eggs in a standard egg carton, which typically holds 12 eggs. Therefore, the possible values for \( X \) are 0, 1, 2, ..., 12, depending on how many eggs are unbroken. Since these are countable values, \( X \) is a discrete random variable.
02

Analyze the Random Variable Y (number of absent students)

The variable \( Y \) represents the number of students on a class list who are absent. The possible values for \( Y \) range from 0 to the total number of students enrolled in the class. Since the number of students is countable, \( Y \) is a discrete random variable.
03

Analyze the Random Variable U (number of swings)

The variable \( U \) is the number of swings a golfer takes before hitting the ball. Possible values start from 1 and go upwards, as a person must swing at least once. The values are countable, making \( U \) a discrete random variable.
04

Analyze the Random Variable X (length of a rattlesnake)

The variable \( X \) signifies the length of a rattlesnake in inches or centimeters. Since the length can take any value within a range (real numbers), \( X \) is a continuous random variable.
05

Analyze the Random Variable Z (amount of royalties earned)

The variable \( Z \) represents royalties from textbook sales. These amounts can vary continuously over a range based on sales quantity and royalty rate. Thus, \( Z \) is a continuous random variable.
06

Analyze the Random Variable Y (pH of soil sample)

The variable \( Y \) stands for the pH level of soil and can take any value between the usual pH range (e.g., 0 to 14). Due to this continuity of values, \( Y \) is a continuous random variable.
07

Analyze the Random Variable X (tension of tennis racket string)

The variable \( X \) is the tension (psi) at which a tennis racket is strung. Tension can take any real value within a specific range, making \( X \) a continuous random variable.
08

Analyze the Random Variable X (total number of coin tosses for a match)

The variable \( X \) represents the number of coin tosses until a match (all heads or all tails) is obtained. The values start from as low as 3 and go upwards indefinitely, which are countable, thus \( X \) is a discrete random variable.

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.

Discrete Random Variables
Discrete random variables are those which can take on a countable number of distinct values. The concept of being countable is key here. For example, consider the number of unbroken eggs in a standard carton holding 12 eggs. Possible values for this variable are limited to integers from 0 to 12, making it countable.
Let's look at another example: the number of students absent on the first day of class. It doesn't matter how many students are enrolled, as long as we can count how many are absent, the variable is discrete.
These types of variables appear when outcomes can be distinctly separated and listed, even if the outcomes are infinite like the number of swings a golfer needs to hit a ball or coin tosses to match heads or tails.
  • Examples include the number of customers entering a store and the number of defective items produced.
  • They typically involve whole numbers as results of counting.
Continuous Random Variables
Continuous random variables differ from discrete variables as they can take on any value within a specified range. This means that there is an infinite number of potential outcomes within any given interval.
An example of this is the length of a rattlesnake, which may be any value within a range, making it a continuous random variable. It fits into the nature of measurements such as length, weight, or time.
Another example is the tension at which a tennis racket is strung. This can vary continuously, within certain bounds, allowing any real number within that range.
  • Examples include temperature, height, and time duration.
  • Values are derived from measurement and can be fractions or decimals.
Probability
The probability of a random variable gives us a way to quantify the likelihood of different outcomes. It's a fundamental concept in both discrete and continuous contexts.
For discrete random variables, the probability function assigns a particular probability to each possible value. For example, the probability of exactly 5 unbroken eggs in a carton.
With continuous random variables, probabilities are usually assigned over intervals (e.g., the probability that a rattlesnake is between 60 and 70 inches long) rather than exact values because individual outcomes have zero probability due to the infinite possibilities between any two points.
  • Probability values range from 0 to 1.
  • A probability of 0 means the event never happens, while 1 means it always happens.
Statistics
Statistics is the science of collecting, analyzing, and interpreting data. It's closely associated with the study of random variables because it helps us understand data based on these variables and their probabilities.
Through statistics, we can estimate parameters, test hypotheses, and make predictions about random variables. For instance, analyzing the distribution of pH values in soil samples helps us understand soil health trends.
Statistics uses concepts like mean, median, and standard deviation to analyze data sets, and it provides methods to deal with uncertainties in data through inferential statistics.
  • Descriptive statistics summarize data sets by measures like average and variability.
  • Inferential statistics focus on making predictions and testing hypotheses using sample data.

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

A manufacturer of flashlight batteries wishes to control the quality of its product by rejecting any lot in which the proportion of batteries having unacceptable voltage appears to be too high. To this end, out of each large lot ( 10,000 batteries), 25 will be selected and tested. If at least 5 of these generate an unacceptable voltage, the entire lot will be rejected. What is the probability that a lot will be rejected if a. Five percent of the batteries in the lot have unacceptable voltages? b. Ten percent of the batteries in the lot have unacceptable voltages? c. Twenty percent of the batteries in the lot have unacceptable voltages? d. What would happen to the probabilities in parts (a)-(c) if the critical rejection number were increased from 5 to 6 ?

An ordinance requiring that a smoke detector be installed in all previously constructed houses has been in effect in a city for 1 year. The fire department is concerned that many houses remain without detectors. Let \(p=\) the true proportion of such houses having detectors, and suppose that a random sample of 25 homes is inspected. If the sample strongly indicates that fewer than \(80 \%\) of all houses have a detector, the fire department will campaign for a mandatory inspection program. Because of the costliness of the program, the department prefers not to call for such inspections unless sample evidence strongly argues for their necessity. Let \(X\) denote the number of homes with detectors among the 25 sampled. Consider rejecting the claim that \(p \geq .8\) if \(x \leq 15\). a. What is the probability that the claim is rejected when the actual value of \(p\) is \(.8\) ? b. What is the probability of not rejecting the claim when \(p=.7\) ? When \(p=.6\) ? c. How do the "error probabilities" of parts (a) and (b) change if the value 15 in the decision rule is replaced by 14 ?

Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses. Suppose this number \(X\) has a Poisson distribution with parameter \(\hat{\lambda}=.2 .(\) Suggested in "Average Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution," J. Qual. Tech., 1983: 126-129.) a. What is the probability that a disk has exactly one missing pulse? b. What is the probability that a disk has at least two missing pulses? c. If two disks are independently selected, what is the probability that neither contains a missing pulse?

a. Use derivatives of the moment generating function to obtain the mean and variance for the Poisson distribution. b. As discussed in Section 3.4, obtain the Poisson mean and variance from \(R_{X}(t)=\ln\) \(\left[M_{X}(t)\right]\). In terms of effort, how does this method compare with the one in part (a)?

The \(n\) candidates for a job have been ranked 1,2 , \(3, \ldots, n\). Let \(X=\) the rank of a randomly selected candidate, so that \(X\) has pmf $$ p(x)=\left\\{\begin{array}{cc} 1 / n & x=1,2,3, \ldots, n \\ 0 & \text { otherwise } \end{array}\right. $$ (this is called the discrete uniform distribution). Compute \(E(X)\) and \(V(X)\) using the shortcut formula. [Hint: The sum of the first \(n\) positive integers is \(n(n+1) / 2\), whereas the sum of their squares is \(n(n+1)(2 n+1) / 6\).]
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.