Discrete Probability Distribution

Oct 13, 2025 by leolavanderia.com.br 141 views

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Introduction

A survey of visitors at a national park was conducted to determine the preferred activity of each visitor to the park. Each visitor chose one activity. The survey results are shown in the table below.

Activity	Number of Visitors
Hiking	120
Camping	80
Swimming	50
Wildlife Watching	30
Photography	20
Other	10

Problem Analysis

The problem is to determine the preferred activity of each visitor to the national park. The survey results show that there are six different activities, and each visitor chose one activity. We need to analyze the data to determine the most popular activity and the least popular activity.

Mathematical Modeling

Let's assume that the number of visitors for each activity is a random variable. We can model the data using a probability distribution. Since the number of visitors for each activity is a discrete random variable, we can use a discrete probability distribution.

Discrete Probability Distribution

A discrete probability distribution is a function that assigns a probability to each possible value of a discrete random variable. In this case, the random variable is the number of visitors for each activity.

Let's define the probability distribution as follows:

P(Hiking) = 120/300 = 0.4
P(Camping) = 80/300 = 0.27
P(Swimming) = 50/300 = 0.17
P(Wildlife Watching) = 30/300 = 0.1
P(Photography) = 20/300 = 0.07
P(Other) = 10/300 = 0.03

Expected Value

The expected value of a discrete random variable is the sum of the product of each possible value and its probability. In this case, the expected value is the average number of visitors for each activity.

Let's calculate the expected value as follows:

E(X) = (120 x 0.4) + (80 x 0.27) + (50 x 0.17) + (30 x 0.1) + (20 x 0.07) + (10 x 0.03) = 48 + 21.6 + 8.5 + 3 + 1.4 + 0.3 = 82.8

Variance

The variance of a discrete random variable is the sum of the product of each possible value and its squared probability. In this case, the variance is the measure of the spread of the data.

Let's calculate the variance as follows:

Var(X) = (120 x 0.4 x 0.4) + (80 x 0.27 x 0.27) + (50 x 0.17 x 0.17) + (30 x 0.1 x 0.1) + (20 x 0.07 x 0.07) + (10 x 0.03 x 0.03) = 19.2 + 7.3 + 2.9 + 0.9 + 0.2 + 0.01 = 30.25

Standard Deviation

The standard deviation of a discrete random variable is the square root of the variance. In this case, the standard deviation is the measure of the spread of the data.

Let's calculate the standard deviation as follows:

SD(X) = sqrt(Var(X)) = sqrt(30.25) = 5.5

Conclusion

In this article, we analyzed the survey results of visitors at a national park to determine the preferred activity of each visitor. We used a discrete probability distribution to model the data and calculated the expected value, variance, and standard deviation. The results show that the most popular activity is hiking, and the least popular activity is other. The standard deviation of 5.5 indicates that the data is spread out, and there is a significant variation in the number of visitors for each activity.

Discussion

The results of this study have implications for park management and planning. By understanding the preferred activities of visitors, park managers can plan and allocate resources more effectively. For example, they can increase the number of hiking trails and campsites to meet the demand for these activities.

Limitations

This study has several limitations. First, the survey results are based on a small sample size, which may not be representative of the entire population of visitors. Second, the survey only asked about the preferred activity of each visitor, and did not collect any additional information about the visitors, such as their demographics or behavior.

Future Research

Future research could build on this study by collecting more data and analyzing it using more advanced statistical techniques. For example, researchers could use regression analysis to examine the relationship between the number of visitors and other factors, such as the time of year or the weather.

References

[1] National Park Service. (2022). Visitor Use Statistics.
[2] [1] National Park Service. (2022). Visitor Use Statistics.

Note: The references are fictional and for demonstration purposes only.

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Introduction

In our previous article, we analyzed the survey results of visitors at a national park to determine the preferred activity of each visitor. We used a discrete probability distribution to model the data and calculated the expected value, variance, and standard deviation. In this article, we will answer some frequently asked questions (FAQs) about the survey results and the analysis.

Q&A

Q: What is the most popular activity among visitors to the national park?

A: The most popular activity among visitors to the national park is hiking, with 120 visitors out of a total of 300.

Q: What is the least popular activity among visitors to the national park?

A: The least popular activity among visitors to the national park is other, with 10 visitors out of a total of 300.

Q: What is the expected value of the number of visitors for each activity?

A: The expected value of the number of visitors for each activity is 82.8.

Q: What is the variance of the number of visitors for each activity?

A: The variance of the number of visitors for each activity is 30.25.

Q: What is the standard deviation of the number of visitors for each activity?

A: The standard deviation of the number of visitors for each activity is 5.5.

Q: Why is the standard deviation of 5.5 significant?

A: The standard deviation of 5.5 is significant because it indicates that the data is spread out, and there is a significant variation in the number of visitors for each activity.

Q: What are the implications of the survey results for park management and planning?

A: The survey results have implications for park management and planning. By understanding the preferred activities of visitors, park managers can plan and allocate resources more effectively. For example, they can increase the number of hiking trails and campsites to meet the demand for these activities.

Q: What are the limitations of the survey results?

A: The survey results have several limitations. First, the survey results are based on a small sample size, which may not be representative of the entire population of visitors. Second, the survey only asked about the preferred activity of each visitor, and did not collect any additional information about the visitors, such as their demographics or behavior.

Q: What are some potential future research directions?

A: Some potential future research directions include collecting more data and analyzing it using more advanced statistical techniques. For example, researchers could use regression analysis to examine the relationship between the number of visitors and other factors, such as the time of year or the weather.

Conclusion

In this article, we answered some frequently asked questions about the survey results and the analysis. We hope that this Q&A guide has provided a better understanding of the survey results and the implications for park management and planning.

Discussion

The survey results have significant implications for park management and planning. By understanding the preferred activities of visitors, park managers can plan and allocate resources more effectively. For example, they can increase the number of hiking trails and campsites to meet the demand for these activities.

Limitations

The survey results have several limitations. First, the survey results are based on a small sample size, which may not be representative of the entire population of visitors. Second, the survey only asked about the preferred activity of each visitor, and did not collect any additional information about the visitors, such as their demographics or behavior.

Future Research

References

[1] National Park Service. (2022). Visitor Use Statistics.
[2] [1] National Park Service. (2022). Visitor Use Statistics.

Note: The references are fictional and for demonstration purposes only.