2. During my office hours (from 12pm to 6pm), an average of about 1 student per hour comes in for help. Assuming that the probability of a student coming in is uniform throughout my office hours, what are the odds that on a particular day I would have 6 students come in for help? What is the mean number of students that will come see me on a particular day? Variance? Standard deviation?
2. During my office hours (from 12pm to 6pm), an average of about 1 student per hour comes in for help. Assuming that the probability of a student coming in is uniform throughout my office hours, what are the odds that on a particular day I would have 6 students come in for help? What is the mean number of students that will come see me on a particular day? Variance? Standard deviation?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**All calculations should be shown by hand**
![**Problem 2: Probability and Statistics Analysis**
During my office hours (from 12 pm to 6 pm), an average of 1 student per hour comes in for help. Assuming that the probability of a student coming in is uniform throughout my office hours, what are the odds that on a particular day I would have 6 students come in for help? What is the mean number of students that will come see me on a particular day? Variance? Standard deviation?
---
**Explanation:**
This problem involves calculating the odds, mean, variance, and standard deviation of a given event within a set timeframe, assuming a uniform distribution of student visits. It requires knowledge of Poisson distribution due to the nature of the events happening during a fixed interval with a known average rate.
1. **Mean Calculation:**
- The mean number of students is the average rate (1 student per hour) multiplied by the number of hours (6 hours), which is 6 students.
2. **Variance and Standard Deviation:**
- In a Poisson distribution, the mean and the variance are equal. Therefore, both are 6 in this context.
- Standard deviation is the square root of variance, which is \(\sqrt{6}\).
3. **Odds of 6 Students Coming:**
- Use the Poisson probability formula:
\[
P(X = k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!}
\]
where \(\lambda\) is the average rate (mean), \(k = 6\), and \(e\) is approximately 2.71828.
This problem provides practice in applying foundational statistical concepts to real-world scenarios, like managing office hours effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8548eebc-136a-4c70-8601-04e6be8c37eb%2Ff0702d07-b1dc-4170-ab5f-9885240afd9f%2Fv2sjdgh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2: Probability and Statistics Analysis**
During my office hours (from 12 pm to 6 pm), an average of 1 student per hour comes in for help. Assuming that the probability of a student coming in is uniform throughout my office hours, what are the odds that on a particular day I would have 6 students come in for help? What is the mean number of students that will come see me on a particular day? Variance? Standard deviation?
---
**Explanation:**
This problem involves calculating the odds, mean, variance, and standard deviation of a given event within a set timeframe, assuming a uniform distribution of student visits. It requires knowledge of Poisson distribution due to the nature of the events happening during a fixed interval with a known average rate.
1. **Mean Calculation:**
- The mean number of students is the average rate (1 student per hour) multiplied by the number of hours (6 hours), which is 6 students.
2. **Variance and Standard Deviation:**
- In a Poisson distribution, the mean and the variance are equal. Therefore, both are 6 in this context.
- Standard deviation is the square root of variance, which is \(\sqrt{6}\).
3. **Odds of 6 Students Coming:**
- Use the Poisson probability formula:
\[
P(X = k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!}
\]
where \(\lambda\) is the average rate (mean), \(k = 6\), and \(e\) is approximately 2.71828.
This problem provides practice in applying foundational statistical concepts to real-world scenarios, like managing office hours effectively.
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