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Chapter 4: Problem 13

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time. Define the random variable \(X\).

Short Answer

Expert verified
\(X\) is the number of events Javier volunteers for, with values 0-5.

Step by step solution

01

Understand Random Variable

A random variable represents numerical outcomes of a random phenomenon. In this context, the random variable \(X\) measures the number of community events Javier volunteers for in a given month.
02

Define the Range of X

The possible values of \(X\) are numerical and correspond to the number of events Javier can attend in a month, which are \(0, 1, 2, 3, 4,\) and \(5\), based on the task description that he does not attend more than five events.
03

Associate Probabilities with Each Outcome

According to the information given, assign probabilities to each value of \(X\): - \(P(X = 0) = 0.05\)- \(P(X = 1) = 0.05\)- \(P(X = 2) = 0.10\)- \(P(X = 3) = 0.20\)- \(P(X = 4) = 0.25\)- \(P(X = 5) = 0.35\).
04

Summarize the Random Variable

The random variable \(X\) is defined as the number of community events Javier volunteers for in one month, where \(X\) takes values from the set \(\{0, 1, 2, 3, 4, 5\}\), with assigned probabilities based on Javier's attending frequency as given.

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Key Concepts

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

Probability Distribution
Probability distribution is a statistical function that describes the likelihood of different outcomes for a random variable. In the case of Javier and his volunteer work at community events, the probability distribution provides a way to see the pattern of his volunteering habits.
It defines how the probabilities are spread out over the possible outcomes of attending different numbers of events.

For instance, in Javier's scenario, the probability distribution is defined as:
  • Attending no events: probability of 0.05
  • Attending one event: probability of 0.05
  • Attending two events: probability of 0.10
  • Attending three events: probability of 0.20
  • Attending four events: probability of 0.25
  • Attending five events: probability of 0.35
Each probability tells us how likely it is for Javier to volunteer at a certain number of events within a month.
This pattern helps to predict and understand Javier's volunteering behavior better.
Numerical Outcomes
Numerical outcomes are possible numeric values that a random variable can assume. When we look at Javier's community events volunteering, each outcome is a count of the events he attends per month.
These outcomes are reflected in the range of the random variable, which, in Javier's case, varies from 0 to 5, representing the number of events he could attend.

Numerical outcomes can be listed as follows:
  • 0 events
  • 1 event
  • 2 events
  • 3 events
  • 4 events
  • 5 events
Each outcome number signifies a distinct possible scenario, showcasing how many events Javier might attend.
This helps form the structure of the random variable and, when paired with their corresponding probabilities, completes the probability distribution.
Community Events
Community events are gatherings or activities that are often organized to support, engage, or bring together members of a community. In Javier's situation, these events are the venues where he volunteers.
Volunteering at such community events can include various tasks like organizing, assisting, or participating in activities centered around community goals.

Javier鈥檚 participation in these community events is also a way of contributing to society. The involvement in community events is not only beneficial for the community, but it also provides Javier with a sense of purpose and belonging. The number of events he attends is the measure of his monthly engagement and is the key factor in determining the random variable in this exercise.
In summary, community events play a significant role in providing Javier with opportunities to volunteer, thereby creating the central focus for defining his random variable and its distribution.

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Most popular questions from this chapter

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next, so that she can plan what classes to offer. Over the years, she has established the following probability distribution. \(\bullet\) Let \(X=\) the number of years a student will study ballet with the teacher. \(\bullet\) Let \(P(x)=\) the probability that a student will study ballet \(x\) years. In words, define the random variable \(X\).

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have established the following probability distribution. Let \(X=\) the number of years a new hire will stay with the company. Let \(P(x)=\) the probability that a new hire will stay with the company \(x\) years. What does the column \(" P(x) "\) sum to?

Use the following information to answer the next four exercises: Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random. Define the random variable \(X\).

A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call. a. In words, define the random variable \(X.\) b. List the values that \(X\) may take on. c. Give the distribution of \(X . X \sim\) ____ (__,___) d. On average, how many dealerships would we expect her to have to call until she finds one that has the car? e. Find the probability that she must call at most four dealerships. f. Find the probability that she must call three or four dealerships.

In one of its Spring catalogs, L.L. Bean庐 advertised footwear on 29 of its 192 catalog pages. Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise footwear. Each page may be picked at most once.. a. In words, define the random variable X. b. List the values that X may take on. C. Give the distribution of \(X . X \sim\) ____ (___,____) d. How many pages do you expect to advertise footwear on them? e. Calculate the standard deviation.
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