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The Poisson distribution has been used by traffic engineers as a model for light traffic, based on the rationale that if the rate is approximately constant and the traffic is light (so the individual cars move independently of each other), the distribution of counts of cars in a given time interval or space area should be nearly Poisson (Gerlough and Schuhl 1955 ). The following table shows the number of right turns during 300 3-min intervals at a specific intersection. Fit a Poisson distribution. Comment on the fit by comparing observed and expected counts. It is useful to know that the 300 intervals were distributed over various hours of the day and various days of the week. $$\begin{array}{cc} \hline n & \text { Frequency } \\ \hline 0 & 14 \\ 1 & 30 \\ 2 & 36 \\ 3 & 68 \\ 4 & 43 \\ 5 & 43 \\ 6 & 30 \\ 7 & 14 \\ 8 & 10 \\ 9 & 6 \\ 10 & 4 \\ 11 & 1 \\ 12 & 1 \\ 13+ & 0 \\ \hline \end{array}$$

Step by step solution

01

Calculate the Mean (λ)

To fit a Poisson distribution, we need to determine the parameter \( \lambda \) (the mean rate of occurrence). Calculate the mean number of turns \( \lambda \) using the formula: \( \lambda = \frac{\sum (n \times \text{Frequency})}{\text{Total Intervals}} \).

02

Compute Expected Frequencies

Once \( \lambda \) is determined, calculate the expected frequency for each \( n \) using the Poisson formula: \( P(n) = \frac{e^{-\lambda} \lambda^n}{n!} \times 300 \). This will give the expected number of intervals for each \( n \).

03

Compare Observed with Expected Counts

Compare the observed frequencies with the expected frequencies by creating a table of the observed values and the calculated expected values for each number of right turns \( n \).

04

Evaluate the Fit

Assess the goodness of the fit by checking how close the observed frequencies are to the expected frequencies. Use statistical tests like Chi-square test if necessary for a more rigorous comparison.

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Key Concepts

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

Traffic Modeling

Traffic modeling is a crucial part of urban planning and civil engineering. It helps understand how vehicles move and how their interactions can be predicted and managed efficiently. The Poisson distribution is often chosen for modeling light traffic scenarios. This distribution is effective because it assumes that events happen independently in a given interval, just like cars making turns at intersections during low traffic times. In the context of the exercise, the distribution helps predict the number of cars making right turns in 3-minute intervals. Each vehicle's decision to turn does not affect others, fitting well with the assumptions of the Poisson process. By studying these patterns in traffic modeling, planners can optimize signal timings and improve road safety, contributing to smoother traffic flow.

Goodness of Fit

The concept of goodness of fit is used to determine how well a statistical model matches observed data. In our exercise, after fitting a Poisson distribution to the turn counts, we need to evaluate if this model properly represents the actual traffic data gathered. A common approach is to compare observed frequencies of right turns with those predicted by the Poisson model. The goodness of fit is determined by how closely these two sets of data align. If the differences between observed and expected values are minor, it indicates a good fit. However, if there are substantial discrepancies, it suggests that the Poisson distribution might not be suitable for this data. This evaluation helps to ensure the reliability and applicability of the model in real-world scenarios.

Mean Rate Calculation

Calculating the mean rate, denoted as \( \lambda \), is the first crucial step in fitting a Poisson distribution to data. The mean rate represents the average number of occurrences in a fixed interval—in this case, the average number of right turns during the 3-minute intervals. To compute \( \lambda \), sum the product of each number of turns \( n \) and its frequency, then divide by the total number of intervals.The formula is: \( \lambda = \frac{\sum (n \times \text{Frequency})}{\text{Total Intervals}} \). This mean rate is the key parameter that helps us model the Poisson distribution. A precise calculation of \( \lambda \) ensures that our expected frequencies are accurate, providing a solid foundation for analyzing the traffic data.

Statistical Testing

Statistical testing is essential for evaluating how well our Poisson distribution fits the observed traffic data. The Chi-square test is often used in this context to statistically check the goodness of the fit between observed and expected frequencies. The test involves calculating the Chi-square statistic, which quantifies the discrepancy between observed and expected values:\[ \chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}} \] If the Chi-square statistic is smaller than a certain critical value (determined from a Chi-square distribution table), it suggests that the observed frequencies do not deviate significantly from what the Poisson model predicts. This statistical testing provides a rigorous method to confirm our initial observations and helps identify any peculiarities in the data that might need closer inspection.