5. The undergraduate library at State U tends to have 33.4 students enter every hour during normal opera (a) Over the next four hours, what is the probability at least 140 students arrive at the library? (b) What is the probability between 10 minutes and 15 minutes pass between consecutive arrivals?
5. The undergraduate library at State U tends to have 33.4 students enter every hour during normal opera (a) Over the next four hours, what is the probability at least 140 students arrive at the library? (b) What is the probability between 10 minutes and 15 minutes pass between consecutive arrivals?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Question 5
Please solve the problem with simple probability rules (no Excel)
![### Probability Problems Related to Student Arrivals at the Library
#### Problem Statement:
The undergraduate library at State U tends to have 33.4 students entering every hour during normal operation.
#### Questions:
(a) Over the next four hours, what is the probability that at least 140 students will arrive at the library?
(b) What is the probability that the time between consecutive arrivals is between 10 minutes and 15 minutes?
#### Explanation and Approach:
In order to address these questions, it's important to have a foundational understanding of Poisson processes and exponential distribution, as these concepts frequently apply to situations involving the timing of events.
**(a) Calculating the Probability of At Least 140 Students Arriving in Four Hours:**
First, figure out the average number of students arriving over four hours. Given 33.4 students per hour:
\[ \lambda = 33.4 \times 4 = 133.6 \]
Here, \(\lambda\) represents the mean number of arrivals in a given time period. In this case, 133.6 is the average student arrival rate over four hours. The number of arrivals in four hours can be modeled as a Poisson distribution \(X \sim \text{Poisson}(133.6)\).
To find out the probability of having at least 140 students arriving, the task is to calculate:
\[ P(X \geq 140) \]
For Poisson distributions, this usually involves either direct calculation using Poisson probability formulas or applying statistical software to compute the exact values.
**(b) Calculating the Probability Between 10 Minutes and 15 Minutes Passes Between Consecutive Arrivals:**
Arrivals in a Poisson process are often modeled using the exponential distribution for the time between consecutive arrivals. The exponential distribution's parameter is:
\[ \lambda = 33.4 \]
To convert \(\lambda\) into minutes:
\[ \lambda = \frac{33.4}{60} \approx 0.5567 \text{ students per minute} \]
The exponential distribution with the rate parameter \(\lambda\) describes the probability \(P(T)\) of the time between arrivals being within a certain range. To find the probability between 10 minutes and 15 minutes:
\[ P(10 \leq T \leq 15) = P(T \leq 15) - P(T \leq 10) \]
Using the cumulative distribution function (](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d41a808-d0ca-407a-8825-29dbf8dd80d4%2Fb3039d1c-f332-451e-a538-7406ba90b227%2Fht8v799_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Probability Problems Related to Student Arrivals at the Library
#### Problem Statement:
The undergraduate library at State U tends to have 33.4 students entering every hour during normal operation.
#### Questions:
(a) Over the next four hours, what is the probability that at least 140 students will arrive at the library?
(b) What is the probability that the time between consecutive arrivals is between 10 minutes and 15 minutes?
#### Explanation and Approach:
In order to address these questions, it's important to have a foundational understanding of Poisson processes and exponential distribution, as these concepts frequently apply to situations involving the timing of events.
**(a) Calculating the Probability of At Least 140 Students Arriving in Four Hours:**
First, figure out the average number of students arriving over four hours. Given 33.4 students per hour:
\[ \lambda = 33.4 \times 4 = 133.6 \]
Here, \(\lambda\) represents the mean number of arrivals in a given time period. In this case, 133.6 is the average student arrival rate over four hours. The number of arrivals in four hours can be modeled as a Poisson distribution \(X \sim \text{Poisson}(133.6)\).
To find out the probability of having at least 140 students arriving, the task is to calculate:
\[ P(X \geq 140) \]
For Poisson distributions, this usually involves either direct calculation using Poisson probability formulas or applying statistical software to compute the exact values.
**(b) Calculating the Probability Between 10 Minutes and 15 Minutes Passes Between Consecutive Arrivals:**
Arrivals in a Poisson process are often modeled using the exponential distribution for the time between consecutive arrivals. The exponential distribution's parameter is:
\[ \lambda = 33.4 \]
To convert \(\lambda\) into minutes:
\[ \lambda = \frac{33.4}{60} \approx 0.5567 \text{ students per minute} \]
The exponential distribution with the rate parameter \(\lambda\) describes the probability \(P(T)\) of the time between arrivals being within a certain range. To find the probability between 10 minutes and 15 minutes:
\[ P(10 \leq T \leq 15) = P(T \leq 15) - P(T \leq 10) \]
Using the cumulative distribution function (
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman